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Research ArticleNew Inner Product Quasilinear Spaces on Interval Numbers

Hacer Bozkurt1 and YJlmaz Yilmaz2

1Department of Mathematics Batman University 72100 Batman Turkey2Department of Mathematics Inonu University 44280 Malatya Turkey

Correspondence should be addressed to Hacer Bozkurt hacerbozkurtbatmanedutr

Received 22 January 2016 Accepted 23 March 2016

Academic Editor Adrian Petrusel

Copyright copy 2016 H Bozkurt and Y Yilmaz This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Primarily we examine the new example of quasilinear spaces namely ldquo119868R119899 interval spacerdquo We obtain some new theorems andresults related to this new quasilinear space After giving some new notions of quasilinear dependence-independence and basison quasilinear functional analysis we obtain some results on 119868R119899 interval space related to these concepts Secondly we present119868119904 119868119888

0 119868119897

infin and 119868119897

2quasilinear spaces and we research some algebraic properties of these spaces We obtain some new results and

provide an important contribution to the improvement of quasilinear functional analysis

1 Introduction

In 1986 Aseev generalized the notion linear spaces byintroducing firmly quasilinear spaces He used the partialorder relation when he defined quasilinear spaces and thenhe can give consistent counterpart of results in linear spacesFor more details the reader can refer to [1] This work hasmotivated a lot of authors to introduce new results on set-valued analysis [2 3]

One of the most useful examples of a quasilinear spaceis the set Ω

119862(119864) of all convex compact subsets of a normed

space 119864 The investigation of this class involves a convexinterval analysis Intervals are excellent tools for handlingglobal optimization problems and for supplementing stan-dard techniques This is because an interval is an infinite setand is thus a carrier of an infinite amount of informationwhich means global information Further the theory of setdifferential equations also needs the analysis ofΩ

119862(119864) [3]

Inspired and motivated by research going on in this areawe introduce an inner product quasilinear space which isdefined in [4] Generally in [4] we give some examplesof quasi-inner product properties on Ω

119862(119864) of all convex

compact subsets of a normed space119864 In our present paperweexamine a new type of a quasilinear space namely 119868R119899 Wefind some important results related to the geometric structureof the 119868R119899 quasilinear space and we examine some algebraic

properties of the 119868R119899 interval space Furthermore we explorethe concepts of 119868119904 119868119888

0 119868119897

infin and 119868119897

2interval sequence spaces

as new examples of quasilinear spaces Moreover we obtainsome theorems and results related to these new spaces whichprovide us with improving the elements of the quasilinearfunctional analysis

2 Quasilinear Spaces and HilbertQuasilinear Spaces

Let us start this section by introducing the definition of aquasilinear space and some of its basic properties given byAseev [1]

Definition 1 A set 119883 is called a quasilinear space if a partialorder relation ldquolerdquo an algebraic sum operation and anoperation of multiplication by real numbers are defined init in such way that the following conditions hold for anyelements 119909 119910 119911 V isin 119883 and any real numbers 120572 120573 isin 1198641

(1) 119909 le 119909(2) 119909 le 119911 if 119909 le 119910 and 119910 le 119911(3) 119909 = 119910 if 119909 le 119910 and 119910 le 119909(4) 119909 + 119910 = 119910 + 119909(5) 119909 + (119910 + 119911) = (119909 + 119910) + 119911

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 2619271 9 pageshttpdxdoiorg10115520162619271

2 Journal of Function Spaces

(6) there exists an element 120579 isin 119883 such that 119909 + 120579 = 119909(7) 120572 sdot (120573 sdot 119909) = (120572 sdot 120573) sdot 119909(8) 120572 sdot (119909 + 119910) = 120572 sdot 119909 + 120572 sdot 119910(9) 1 sdot 119909 = 119909(10) 0 sdot 119909 = 120579(11) (120572 + 120573) sdot 119909 le 120572 sdot 119909 + 120573 sdot 119909(12) 119909 + 119911 le 119910 + V if 119909 le 119910 and 119911 le V(13) 120572 sdot 119909 le 120572 sdot 119910 if 119909 le 119910

A linear space is a quasilinear space with the partial orderrelation ldquo=rdquo The most popular example which is not a linearspace is the set of all closed intervals of real numbers with theinclusion relation ldquosuberdquo algebraic sum operation

119860 + 119861 = 119886 + 119887 119886 isin 119860 119887 isin 119861 (1)

and the real-scalar multiplication

120582119860 = 120582119886 119886 isin 119860 (2)

We denote this set byΩ119862(R) Another one isΩ(R) the set of

all compact subsets of real numbers By a slight modificationof algebraic sum operation (with closure) such as

119860 + 119861 = 119886 + 119887 119886 isin 119860 119887 isin 119861 (3)

and by the same real-scalar multiplication defined above andby the inclusion relation we get the nonlinear QLS Ω

119862(119864)

and Ω(119864) the space of all nonempty closed bounded andconvex closed bounded subsets of some normed linear space119864 respectively

Lemma 2 Suppose that any element 119909 in a QLS 119883 has aninverse element 1199091015840 isin 119883 Then the partial order in 119883 is deter-mined by equality the distributivity conditions hold andconsequently119883 is a linear space [1]

Suppose that 119883 is a QLS and 119884 sube 119883 Then 119884 is calleda subspace of 119883 whenever 119884 is a QLS with the same partialorder and the restriction to 119884 of the operations on 119883 Onecan easily prove the following theorem using the condition ofbeing a QLS It is quite similar to its linear space analogue

Theorem 3 119884 is a subspace of a QLS119883 if and only if for every119909 119910 isin 119884 and 120572 120573 isin 119877 120572119909 + 120573119910 isin 119884 [5]

Let 119883 be a QLS An 119909 isin 119883 is said to be symmetric ifminus119909 = 119909 where minus119909 = (minus1)119909 and 119883

119889denotes the set of all

such elements 120579 denotes the zerorsquos additive unit of119883 and it isminimal that is 119909 = 120579 if 119909 le 120579 An element 1199091015840 is called inverseof 119909 if 119909 + 1199091015840 = 120579 The inverse is unique whenever it existsand 1199091015840 = minus119909 in this case Sometimes 1199091015840 may not exist but minus119909is always meaningful in QLSs An element 119909 possessing aninverse is called regular otherwise it is called singular For asingular element 119909 we should note that 119909 minus 119909 = 0 Now 119883

119903

and 119883119904stand for the sets of all regular and singular elements

in119883 respectively Further119883119903 119883

119889 and119883

119904cup0 are subspaces

of 119883 and they are called regular symmetric and singular

subspaces of119883 respectively [5] It is easy from the definitionsthat119883

119889sub 119883

119904and119883 = 119883

119903cup 119883

119904

In a linearQLS that is in a linear space there is no singularelement Further in a QLS119883 it is obvious that any element 119909is regular if and only if 119909 minus 119909 = 0

Proposition 4 In a QLS119883 if 119909 isin 119883 and 119910 isin 119883119904then 119909+119910 isin

119883119904

Proposition 5 In a quasilinear space119883 every regular elementis minimal [5]

Definition 6 Let119883 be a QLS and 119910 isin 119883 The set of all regularelements preceding 119910 is called floor of 119910 and 119865

119910denotes the

set of all such elements Therefore

119865119910= 119911 isin 119883

119903 119911 le 119910 (4)

The floor of any subset 119872 of 119883 is the union of floors of allelements in119872 and is denoted by 119865

119872[6]

Definition 7 Let 119883 be a QLS 119872 sube 119883 and 119909 119910 isin 119872 119872 iscalled a proper set if the following two conditions hold

(i) 119865119911= 0 for all 119911 isin 119872

(ii) 119865119909= 119865119910for each pair of points 119909 119910 with 119909 = 119910

Otherwise119872 is called an improper set Particularly if 119883is a proper set then it is called a proper quasilinear space(briefly proper qls) [6]

Definition 8 Let119883 be a quasilinear space119883 is called a solid-floored quasilinear space whenever

119910 = sup 119909 isin 119883119903 119909 le 119910 (5)

for each 119910 isin 119883 Otherwise 119883 is called a non-solid-flooredquasilinear space [6]

Definition 9 Let 119883 be a QLS Consolidation of floor of 119883 isthe smallest solid-flooredQLS containing119883 that is if thereexists another solid-floored QLS 119884 containing119883 then sube 119884

Clearly = 119883 for some solid-floored QLS 119883 FurtherΩ119862(R119899

)119904= Ω

119862(R119899

) For a QLS119883 the set

119865

119910= 119911 isin ()

119903

119911 ⪯ 119910 (6)

is the floor of119883 in

Definition 10 Let119883 be a QLS A real function sdot119883 119883 rarr 119864

1

is called a norm if the following conditions hold [1]

(14) 119909119883gt 0 if 119909 = 0

(15) 119909 + 119910119883le 119909

119883+ 119910

119883

(16) 120572119909119883= |120572|119909

119883

(17) if 119909 le 119910 then 119909119883le 119910

119883

(18) if for any 120576 gt 0 there exists an element 119909120576isin 119883 such

that 119909 le 119910 + 119909120576and 119909

120576119883le 120576 then 119909 le 119910

Journal of Function Spaces 3

A quasilinear space 119883 with a norm defined on it is callednormed QLS It follows from Lemma 2 that if any 119909 isin 119883 hasan inverse element 1199091015840 isin 119883 then the concept of a normedQLScoincides with the concept of a real normed linear space

Let119883 be a normed QLS Hausdorff or norm metric on119883is defined by the equality

ℎ119883(119909 119910)

= inf 119903 ge 0 119909 le 119910 + 1198861199031 119910 le 119909 + 119886

119903

21003817100381710038171003817119886

119903

119894

1003817100381710038171003817 le 119903

(7)

Since 119909 le 119910 + (119909 minus 119910) and 119910 le 119909 + (119910 minus 119909) the quantityℎ119883(119909 119910) is well defined for any elements 119909 119910 isin 119883 and

ℎ119883(119909 119910) le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119883 (8)

It is not hard to see that this function ℎ119883(119909 119910) satisfies all of

the metric axioms

Lemma 11 Theoperations of algebraic sum andmultiplicationby real numbers are continuous with respect to the Hausdorffmetric The norm is a continuous function with respect to theHausdorff metric [1]

Example 12 Let 119864 be a Banach space A norm on Ω(119864) isdefined by

119860Ω(119864)= sup

119886isin119860

119886119864 (9)

ThenΩ(119864) andΩ119862(119864) are normed quasilinear spaces In this

case the Hausdorff metric is defined as usual

ℎΩ119862(119864)

(119860 119861)

= inf 119903 ge 0 119860 sub 119861 + 119878119903(120579) 119861 sub 119860 + 119878

119903(120579)

(10)

where 119878119903(120579) denotes a closed ball of radius 119903 about 120579 isin 119883 [1]

Let us give an extended definition of inner-product Thisdefinition and some prerequisites are given by Y Yılmaz Wecan see following inner-product as (set-valued) inner producton QLSs

Definition 13 Let119883 be a quasilinear space A mapping ⟨sdot sdot⟩ 119883 times 119883 rarr Ω(R) is called an inner product on 119883 if for any119909 119910 119911 isin 119883 and 120572 isin R the following conditions are satisfied

(19) if 119909 119910 isin 119883119903then ⟨119909 119910⟩ isin Ω

119862(R)

119903equiv R

(20) ⟨119909 + 119910 119911⟩ sube ⟨119909 119911⟩ + ⟨119910 119911⟩(21) ⟨120572 sdot 119909 119910⟩ = 120572 sdot ⟨119909 119910⟩(22) ⟨119909 119910⟩ = ⟨119910 119909⟩(23) ⟨119909 119909⟩ ge 0 for 119909 isin 119883

119903and ⟨119909 119909⟩ = 0 hArr 119909 = 0

(24) ⟨119909 119910⟩Ω(R) = sup⟨119886 119887⟩Ω(R) 119886 isin 119865

119909 119887 isin 119865

119910

(25) if 119909 ⪯ 119910 and 119906 ⪯ V then ⟨119909 119906⟩ sube ⟨119910 V⟩(26) if for any 120576 gt 0 there exists an element 119909

120576isin 119883 such

that 119909 ⪯ 119910 + 119909120576and ⟨119909

120576 119909

120576⟩ sube 119878

120576(120579) then 119909 ⪯ 119910

A quasilinear space with an inner product is called aninner product quasilinear space briefly IPQLS

Example 14 One can see easily Ω119862(R) the space of closed

real intervals is a IPQLS with inner-product defined by

⟨119860 119861⟩ = 119886 sdot 119887 119886 isin 119860 119887 isin 119861 (11)

Every IPQLS 119883 is a normed QLS with the norm definedby

119909 = radic⟨119909 119909⟩Ω(R) (12)

for every 119909 isin 119883 [4]

Proposition 15 If in an IPQLS 119909119899rarr 119909 and 119910

119899rarr 119910 then

⟨119909119899 119910

119899⟩ rarr ⟨119909 119910⟩ [4]

AN IPQLS is called Hilbert QLS if it is complete accord-ing to the Hausdorff (norm) metric

Example 16 Let 119864 be an inner product space Then we knowthat Ω(119864) is an IPQLS and it is complete with respect to theHausdorff metric SoΩ(119864) is a Hilbert QLS [4]

Definition 17 (orthogonality) An element 119909 of an IPQLS119883 issaid to be orthogonal to an element 119910 isin 119883 if

1003817100381710038171003817⟨119909 119910⟩1003817100381710038171003817Ω(R)

= 0 (13)

We also say that 119909 and 119910 are orthogonal and we write 119909 perp 119910Similarly for subsets119898 119899 sube 119883 we write 119909 perp 119898 if 119909 perp 119911 for all119911 isin 119898 and119898 perp 119899 if 119886 perp 119887 for all 119886 isin 119898 and 119887 isin 119899 [4]

An orthonormal set 119872 sub 119883 is an orthogonal set in 119883whose elements have norm 1 that is for all 119909 119910 isin 119872

1003817100381710038171003817⟨119909 119910⟩1003817100381710038171003817Ω(R)

=

0 119909 = 119910

1 119909 = 119910

(14)

Definition 18 Let119860 be a nonempty subset of an inner productquasilinear space 119883 An element 119909 isin 119883 is said to beorthogonal to 119860 denoted by 119909 perp 119860 if ⟨119909 119910⟩

Ω(R) = 0 forevery 119910 isin 119860 The set of all elements of 119883 orthogonal to 119860denoted by119860perp is called the orthogonal complement of119860 andis indicated by

119860perp

= 119909 isin 119883 1003817100381710038171003817⟨119909 119910⟩

1003817100381710038171003817Ω(R)= 0 119910 isin 119860 (15)

Theorem 19 For any subset 119860 of an IPQLS119883 119860perp is a closedsubspace of119883 [4]

Definition 20 Let119883 be a quasilinear space 119909119896119899

119896=1sub 119883 The

element 119909 isin 119883 with

12057211199091+ 120572

21199092+ sdot sdot sdot + 120572

119899119909119899le 119909 (16)

is said to be a quasilinear combination (ql-combination) of119909

119896119899

119896=1corresponding to scalars 120572

119896119899

119896=1sub R ql-combination

of 119909119896119899

119896=1corresponding to the scalars 120572

119896119899

119896=1may not be

unique from the definition while it is well known that the


linear combination of 119909119896119899

119896=1corresponding to these scalars

is unique The set

Qsp119860 = 119909 isin 119883 119899

sum

119896=1

120572119896119909119896le 119909 119909

1 119909

2 119909

119899

isin 119860 1205721 120572

2 120572

119899isin R

(17)

is said to be quasispan of119860 and is denoted by Qsp119860 One cansee easily that Qsp119860 is a subspace of119883 [7]

It is clear that Sp119860 sube Qsp119860 and Sp119860 = Qsp119860 iff 119883 is alinear space where Sp119860 is the span of 119860 in119883

Definition 21 Let 119883 be a QLS 119909119896119899

119896=1sub 119883 and 120582

119896119899

119896=1sub R

If

120579 le 12058211199091+ 120582

21199092+ sdot sdot sdot + 120582

119899119909119899

(18)

implies 1205821= 120582

2= sdot sdot sdot = 120582

119899= 0 then 119909

119896119899

119896=1is said to

be quasilinear independent (ql-independent) otherwise it issaid to be ql-dependent in119883 [7]

Theorem 22 Any set which has 119899 + 1 elements has to be ql-dependent inΩ

119862(R119899

) [7]

Definition 23 Let 119883 be a QLS and let 119860 be a ql-independentsubset of 119883 If Qsp119860 = 119883 then the set 119860 is called a (Hamel)basis for119883 [7]

Definition 24 Singular dimension of QLS119883 is defined as themaximum number of ql-independent elements in singularsubspace of 119883 If this number is finite then 119883 is called finitesingular dimensional otherwise it is called infinite singulardimensional Further the dimension of regular subspace of119883is called regular dimension of 119883 It is denoted by 119904 minus dim119883and 119903 minus dim119883 respectively [6]

On the other hand if 119904 minus dim119883 = 119903 minus dim119883 = 119886 then 119886is said to be dimension of119883 and is written as dim119883 = 119886 [6]

Definition 25 Let 119883 be a solid-floored quasilinear space ASchauder basis is a sequence (119909

119899) of elements of 119883 such that

for every element 119910 isin 119883 and for every 119911 isin 119865119910there exists a

unique sequence of 120572119899 scalars so that

119911 =

infin

sum

119899=1

120572(119911)

119899119909119899

119910 = sup 119909 isin 119883119903 119909 le 119910

(19)

This definition is identical to the following definition (119909119899) is

a Schauder basis of119883 if and only if

ℎ(119911

119896

sum

119899=1

120572(119911)

119899119909119899) 997888rarr 0 (119896 997888rarr infin)

119910 = sup 119909 isin 119883119903 119909 le 119910

(20)

for every 119910 isin 119883 and for every 119911 isin 119865119910

Example 26 Ω(R) and Ω119862(R) are solid-floored quasilinear

space But singular subspace ofΩ119862(R) is a non-solid-floored

quasilinear space

Recall that the closed interval denoted by [119886 119887] is the setof real numbers given by [119886 119887] = 119909 isin R 119886 le 119909 le

119887 Although various other types of intervals (open half-open) appear throughout mathematics our work will centerprimarily on closed intervals In this paper the term intervalwill mean closed interval

3 Interval Spaces

In this section we introduce some new examples of quasi-linear space Let 119868R be a set of all closed intervals of realnumbers In this case 119868R is the whole compact convex subsetof real numbers Now we can give two-dimensional intervalvector such that 119868R2

= (1198831 119883

2) 119883

1 119883

2isin 119868R And so we

can give

119868R119899

= (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin 119868R (21)

We should state that 119868R119899 is not a vector space that is not linearspace Also the convergence of these sets was discussed in [8]

Example 27 Let119883 = (1198831 119883

2 119883

119899) isin 119868R119899 and119884 = (119884

1 119884

2

119884119899) isin 119868R119899The algebraic sum operation on 119868R119899 is defined

by the expression

119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899) (22)

and multiplication by a real number 120572 isin R is defined by

120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899) (23)

If we will assume that the partial order on 119868R119899 is given by

119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (24)

then 119868R119899 is a quasilinear space with the above sum operationmultiplication and partial order relation From here we getthat 119868R119899 is another example of quasilinear spaces

Remark 28 The quasilinear space 119868R119899 and the quasilinearspace Ω

119862(R119899

) are different from each other For examplewhile the set 119860 = (119909 119910) 1199092 + 1199102 le 1 is an element ofΩ119862(R2

) it is not an element of 119868R2 On the other hand119861 = ([1 2] 2) isin 119868R2 but 119861 notin Ω

119862(R2

) So 119868R119899 and Ω119862(R119899

)

are two different examples of quasilinear spaces

Example 29 Let 119860 = (1 0) sub 119868R2 We can find minus119860 =(minus1 0) such that 119860 + (minus119860) = (0 0) So 119860 is a regularsubset of 119868R2 Also if 119861

119894isin R for every 119861 = (119861

1 119861

2 119861

119899) isin

119868R119899 then 119861 is a regular subset of 119868R119899 On the other handif 119886

119894= 119887119894for 1 le 119894 le 119899 and 119886

119894 119887119894isin R then 119861 = ([119886

1 1198871]

[1198862 1198872] [119886

119899 119887119899]) is a singular subset of 119868R119899

The norm on 119868R119899 is defined by

119883 =1003817100381710038171003817(1198831 1198832 119883119899)

1003817100381710038171003817 = sup1le119894le119899

10038171003817100381710038171198831198941003817100381710038171003817119868R (25)


Then 119868R119899 is a normed quasilinear space From [1] we knowthat 119868R is a normed quasilinear space with 119883 = sup

119909isin119883|119909|

For example let 119860 = ([1 3] [2 3] [minus1 0]) isin 119868R3 Then weget 119860 = sup[1 3]

119868R [2 3]119868R [minus1 0]119868R = 3The Hausdorff metric is defined as usual

ℎ119868R (119860 119861)

= inf 119903 ge 0 119860 sube 119861 + 119878119903(120579) 119861 sube 119860 + 119878

119903(120579)

(26)

where 119878119903(120579) is the closed ball of radius 119903 about 120579 isin 119883

Theorem 30 119868R119899 interval space is a Banach Ω-space with119883 = sup

1le119894le119899119883

119894119868R

Proof Let 119883119899 be any Cauchy sequence in 119868R119899 Then givenany 120598 gt 0 there is119873 such that for all119898 119899 gt 119873 we have

119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(27)

Hence for any 1 le 119894 le 119899 we get 119860119895120598119894 le 120598 On the other hand

since 119868R119899 is a quasilinear space

119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(28)

for every 1 le 119894 le 119899 This shows that 119883119899119894is a Cauchy sequence

in 119868R By completeness of 119868R there is 119883119894isin 119868R such that

119883119899

119894rarr 119883

119894for all 119894 isin [1 119899] Furthermore since 119883

119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883

119899) isin 119868R119899 From (28) by

letting119898 rarrinfin we obtain

119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598 119895 = 1 2

(29)

for all 119899 gt 1198990 So we get 119861119895120598 le 120598 since 119861119895120598 = sup119861119895120598

119894119868R

Also we obtain

119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(30)

for all 1 le 119894 le 119899Thismeans that the sequence (119883119899) convergesto119883 in 119868R119899

Now we will show that 119868R119899 is anΩ-space Suppose 119861119883=

(

119899 units⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞[minus1 1] [minus1 1] [minus1 1]) From the definition of Ω-spacesince

119883 = sup 100381710038171003817100381711988311003817100381710038171003817 100381710038171003817100381711988321003817100381710038171003817

10038171003817100381710038171198831198991003817100381710038171003817 le10038171003817100381710038171198611198831003817100381710038171003817 = 1 (31)

we have 1198831 le 1 119883

2 le 1 119883

119899 le 1 hArr 119883

1sube [minus1 1]

1198832sube [minus1 1] 119883

119899sube [minus1 1] This proves the theorem

Example 31 The space 119868R119899 is a normed quasilinear spacewith norm defined by

1198832= (

119899

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(32)

The quasilinear space 119868R119899 with the inner product

⟨119883 119884⟩ =

119899

sum

119894=1

⟨119883119894 119884

119894⟩119868R (33)

is an inner product quasilinear space

Theorem 32 119868R119899 is a Hilbert quasilinear space with sdot 2

norm

Proof The norm in Example 31 is defined by 1198832=

(sum119899

119894=1119883

1198942

119868R)12 and can be obtained from the inner product

quasilinear in Example 31 Let (119883119899) be any Cauchy sequencein 119868R119899 Then given any 120598 gt 0 there is 119873 such that for all119898 119899 gt 119873 we have

119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(34)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(35)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883

119899) isin 119868R119899 From the above

inequality by letting119898 rarrinfin we obtain

119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

=120598

radic119899 119895 = 1 2

(36)

for all 119899 gt 1198990 So we get

119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(37)

and 119861119895120598 le 120598 for all 1 le 119894 le 119899 This means that the sequence(119883

119899

) is convergent to119883 in 119868R119899

Example 33 Let 119883 = (119868R2

sube) and take the singleton 119860 =([1 2] [minus1 3]) in119883 The q-span of 119860 is

Qsp119860 = 119883 isin 119868R2

120582 sdot ([1 2] [minus1 3]) sube 119883 120582 isin R (38)


For example ([0 2] [minus1 4]) isin Qsp119860 since ([1 2] [minus1 3]) sube([0 2] [minus1 4]) whereas ([minus1 3] [1 2]) notin Qsp119860 Becausethere is no 120582 isin R such that 120582 sdot ([1 2] [minus1 3]) sube ([minus1 3][1 2]) clearly Qsp119860 = 119868R2 Let 119861 = (1 0) (0 1) beanother element of 119883 For any 119909 isin 119883 clearly we can write1205821sdot (1 0) + 120582

2sdot (0 1) sube 119909 for some 120582

1 120582

2isin R This

means Qsp119861 = 119883

Example 34 In (119868R2

sube) let 119860 = ([2 3] [minus1 1]) and 119861 =([minus1 1] [minus1 0])Then119860 is ql-independent since (0 0) sube120582 sdot ([2 3] [minus1 1]) for 120582 = 0 where (0 0) is the zero ofthe quasilinear space 119868R2 In contrast the singleton 119861 is ql-dependent since (0 0) sube 120573 sdot ([minus1 1] [minus1 0]) for 120573 = 1This is an unusual case in linear spaces because a nonzerosingleton is clearly a linearly independent set in such caseParticularly any subset of 119868R2 possessing every vector of anelement related to zero must be ql-dependent even if it is asingleton

Example 35 From Example 33 Qsp119861 = 119868R2 Also 119861 is ql-independent since (0 0) sube 120572

1sdot (1 0) + 120572

2sdot (0 1)

From here 119861 = (1 0) (0 1) is a basis for 119868R2

Corollary 36 119868R2 is a normed quasilinear space with 1198832=

(sum2

119894=1119883

1198942

119868R)12 From Example 31 119868R2 is an inner product

quasilinear space with (33) Further the set (1 0) (01) is an orthonormal subset of 119868R2

Hence (1 0) (01) is an orthonormal basis for 119868R2

Example 37 Let 119860 = ([minus1 1] [0 2]) isin 119868R2

Floor of 119860 is

119865119860= (119910

1 119910

2) isin (119868R

2

)119903

(1199101 119910

2) sube 119860

= (1199101 119910

2) isin R

2

minus1 le 1199101le 1 0 le 119910

2le 2

(39)

Floor of119883 = (1198831 119883

2 119883

119899) isin 119868R119899 is

119865119868R119899 = ⋃119865119883 = ⋃(1199101 1199102 119910119899) isin (119868R

119899

)119903 119910

1

isin 1198831 119910

2isin 119883

2 119910

119899isin 119883

119899

(40)

Theorem 38 The space 119868R119899 is a solid-floored quasilinearspace

Proof Let 119883 isin 119868R119899 We know that119883 = (1198831 119883

2 119883

119899) and

119883119894isin 119868R for all 1 le 119894 le 119899 From the definition of the floor of

an element we have

119865119883= 119884 isin (119868R

119899

)119903 119884 le 119883 (41)

Since 119868R119899 is a quasilinear space if 119884 le 119883 then we find119884119894le 119883

119894for every 1 le 119894 le 119899 Here 119883

119894isin 119868R for every 119883 =

(1198831 119883

2 119883

119899) isin 119868R119899 and 1 le 119894 le 119899 Since 119884 isin R119899

119884119894isin R

for all 119894 isin [1 119899] Hence

119883119894= sup119865

119883119894= sup 119884

119894isin R 119884

119894le 119883

119894 (42)

since 119868R is a solid-floored quasilinear space This shows that119883 = sup119865

119883for every119883 = (119883

1 119883

2 119883

119899) isin 119868R119899

Example 39 The set 119882 = 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2

119883119899isin (119868R)

119889 consists of all symmetric elements of 119868R119899

119882 is a non-solid-floored quasilinear space since supremumof floors of every element in119882 is 0 For the same reasonthe subspace of 119868R119899

119881

= 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119900119889

(43)

is non-solid-floored

Lemma 40 The space 119868R119899 is a proper quasilinear space

Proof Let us take arbitrary elements 119860 =

(1198601 119860

2 119860

119899) 119861 = (119861

1 119861

2 119861

119899) isin 119868R119899 such that

119860 = 119861 If 119860 ⊊ 119861 then 119860119894⊊ 119861

119894for at least 119894 isin [1 119899] Hence

there is at least 119886119894isin 119860

119894such that 119886

119894notin 119861

119894 So 119886

119894 sube 119860

119894and

119886119894 ⊊ 119861

119894 From here we have

(1198861 119886

2 119886

119899) sube (119860

1 119860

2 119860

119899)

(1198861 119886

2 119886

119899) ⊊ (119861

1 119861

2 119861

119899)

(44)

If 119861 ⊊ 119860 then 119861119894⊊ 119860

119894for at least 119894 isin [1 119899] Hence there is at

least 119887119894isin 119861

119894such that 119887

119894notin 119860

119894 So 119887

119894 sube 119861

119894and 119887

119894 ⊊ 119860

119894 From

here we have (1198871 119887

2 119887

119899) sube (119861

1 119861

2 119861

119899) = 119861

and (1198871 119887

2 119887

119899) ⊊ (119860

1 119860

2 119860

119899) = 119860 If there

is not a comparison between 119860 and 119861 then there is not acomparison between 119860

119894and 119861

119894for some 119894 isin [1 119899] Hence

there exist two elements 119886119894isin 119860

119894 119886

119894notin 119861

119894and 119887

119894isin 119861

119894

119887119894notin 119860

119894for some 119894 isin [1 119899] Thus 119886

119894 sube 119860

119894 119886

119894 ⊊

119861119894and 119887

119894 sube 119861

119894 119887

119894 ⊊ 119860

119894for this 119894 isin [1 119899] So we

have (1198861 119886

2 119886

119899) sube 119860 (119886

1 119886

2 119886

119899) ⊊ 119861 and

(1198871 119887

2 119887

119899) sube 119861 (119887

1 119887

2 119887

119899) ⊊ 119860 So 119868R119899 is

a proper quasilinear space

Example 41 The singular subspace of 119868R119899

(119868R119899

)119904

= 0

cup (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119904

(45)

is improper because in this space floors of some elementsmay be empty set and floors of any two different elementsmaybe the same

Example 42 Regular and singular dimensions of the quasi-linear spaces R119899

119868R119899

(119868R119899

)119904 and (119868R119899

)119903are as follows

119903 minus dim (R119899

) = 119899

119904 minus dim (R119899

) = 0

119903 minus dim (119868R119899

) = 119899

119904 minus dim (119868R119899

) = 119899

119903 minus dim (119868R119899

)119904= 0

119904 minus dim (119868R119899

)119904= 119899


119903 minus dim (119868R119899

)119903= 119899

119904 minus dim (119868R119899

)119903= 0

(46)

Example 43 Let us consider the subspace 119860 = (119868R2

)119904cup

(119905 0) 119905 isin R of 119868R2 and the elements 1198861= ([1 2] 0)

and 1198862= (0 [2 4]) of 119860

119904 The set 119886

1 1198862 is ql-independent

in 119860119904since there are nonzero scalars 120582

1and 120582

2satisfying the

inclusion (0 0) sube 12058211198861+ 120582

21198862 Hence singular dimension

of119860must be greater than or equal to 2 Remember that119860 is asubspace of 119868R2 andTheorem 22Then 119904minusdim119860 = 2Clearly119860119903is equivalent to R and so 119903 minus dim119860 = 1

4 Interval Sequence Spaces

In this section we will introduce interval sequence spaces asa different example of quasilinear spaces We first define thefollowing new interval spaces The set

119868119904 = 119883 = (119883119899) isin 119868R

infin

(119883119899) = (119883

1 119883

2 119883

119899 )

isin 119868Rinfin

(47)

is called all intervals sequence space Let 119883 = (1198831 119883

2

119883119899) isin 119868R119899 and 119884 = (119884

1 119884

2 119884

119899) isin 119868R119899 The algebraic sum

operation on 119868119904 is defined by the expression

119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899 ) (48)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899 ) (49)

If we will assume that the partial order on 119868119904 is given by

119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (50)

then 119868119904 is a quasilinear space with the above sum operationmultiplication and partial order relation 119868119904 is a metric spacewith

119889 (119883 119884) =

infin

sum

119894=1

1

2119894[ℎ (119883

119894 119884

119894)

1 + ℎ (119883119894 119884

119894)] (51)

for every 119883119884 isin 119868119904 Here ℎ is the known Hausdorff metricon 119868R

Now we give a new interval sequence space denoted by119868119897infin

119868119897infin= 119883 = (119883

119899) isin 119868R

infin

(119883119899) bounded (52)

namely the space of all bounded interval sequences (119883119899) of

real numbers such that

sup 10038171003817100381710038171198831198941003817100381710038171003817119868R le infin lArrrArr 119883 isin 119868119897

infin (53)

Here the boundedness of interval sequences will be consid-ered according to the Hausdorff metric 119868119897

infinis a quasilinear

space with (48) (49) and (50)The sum operation and scalarmultiplication are well defined since

sup 1003817100381710038171003817119883119894 + 1198841198941003817100381710038171003817119868R le sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R+10038171003817100381710038171198841198941003817100381710038171003817119868R

le sup 10038171003817100381710038171198831198941003817100381710038171003817119868R + sup 1003817100381710038171003817119884119894

1003817100381710038171003817119868R

le infin

sup 10038171003817100381710038171205821198831198941003817100381710038171003817119868R le sup 120582 1003817100381710038171003817119883119894

1003817100381710038171003817119868R le 120582 sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R

le infin

(54)

119868119897infin

is a normed quasilinear space with norm functiondefined by

119883 = sup 10038171003817100381710038171198831198991003817100381710038171003817119868R 119899 isin N (55)

Indeed since it is easy to verify conditions (14)ndash(17) we onlyprove condition (18) Let 119883 le 119884 + 119883

120598and 119883

120598 le 120598 for every

119883119884 isin 119868119897infin Then we have 119883

120598119894119868R le 120598 for all 1 le 119894 le infin On

the other hand we find119883119894le 119884

119894+ 119883

120598119894for all 1 le 119894 le infin when

119883 le 119884 + 119883120598 Since 119868R is a normed quasilinear space we get

119883119894le 119884

119894for every 119894 isin [1infin] Thus we obtain119883 le 119884

Denote

1198681198880= 119883 = (119883

119899) isin 119868R

infin

(119883119899) 997888rarr 0 (56)

that is the space 1198681198880consists of all convergent sequences (119883

119899)

whose limit is zero such that

lim119899rarrinfin

10038171003817100381710038171198831198991003817100381710038171003817119868R= 0 lArrrArr 119883 isin 119868119888

0 (57)

1198681198880is a quasilinear space the same as 119868119897

infinwith (48) (49) and

(50)The space 119868119888

0with the norm

119883 = sup119894isinN

10038171003817100381710038171198831198941003817100381710038171003817119868R (58)

is a normed quasilinear spaceA new example of the nonlinear quasilinear space is

defined by

1198681198972= 119883 = (119883

119899) isin 119868R

infin

infin

sum

119894=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868Rbounded (59)

1198681198972is a not linear space

Example 44 1198681198972is a metric space with

119889 (119883 119884)

= inf 119903 ge 0 119883 sube 119884 + 119878119903(120579) 119884 sube 119883 + 119878

119903(120579)

(60)

where 119878119903(120579) = 119883 isin 119868119897

2 119883 le 119903

Proposition 45 1198681198972with the norm defined by

119883 = (

infin

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(61)

is a normed quasilinear space but 1198681198972is not a Banach quasilin-

ear space with (61)


Proof First we show that 1198681198972is a quasilinear space then that

(61) is a norm in 1198681198972 and finally that the space is not complete

The fact that 1198681198972is a quasilinear space with (48) (49) and (50)

is obvious Here sum operation and scaler multiplication arewell defined since

(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)

for every 119883119884 isin 1198681198972and 120582 isin R Also (61) provides the (14)ndash

(18) conditions and 1198681198972is a normed quasilinear space with

(61)Now we assume that119883119899 is a Cauchy sequence in 119868119897

2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)

then given any 120598 gt 0 there exists a number 1198990such that

119889 (119883119899

119894 119883

119898

119894) le 120598 (64)

Namely for every 120598 gt 0 there exists a number 1198990such that

119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)

Note that this implies that for every fixed 119894 isin N and for every120598 gt 0 there exists a number 119899

0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)

But this means that for every 119894 the sequence (119883119899119894) is a Cauchy

sequence in 119868R and thus convergent Denote

119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)

We are not going to prove that119883 is an element of 1198681198972and that

the sequence (119883119899) does not converge to119883 Indeed from (65)by letting119898 rarrinfin we obtain

119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)

for every 119899 ge 1198990 Here for every 119894 we get

119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)

So we have that 119883119899 is not convergent to 119883 whether or notsuminfin

119894=1119883

1198942

le infin

Corollary 46 1198681198972interval sequence space is not a Banach

quasilinear space with (61) since

119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897

infininterval sequence spaces are

solid-floored quasilinear space Definition of the 1198681198972interval

sequence space 119883 = (119883119899) isin 119868119897

2hArr sum

infin

119899=1119883

1198992

119868R lt infin Wefind definition of solid-flooredness in which

119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)

Since 1198681198972is a quasilinear space we find 119884

119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)

for every 1 le 119894 lt infin Here we see clearly that 119884119894isin 119865

119883119894

Otherwise since 119868R is a solid-floored quasilinear space wehave

119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)

for every 119894 Hence we obtain 119883 = sup119865119883

since 1198681198972is a

quasilinear space with ldquolerdquo relation

Example 48 Singular subspace of 1198681198880is a non-solid-floored

quasilinear space For example we have sup119865119861= (0 0 ) =

0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904

Example 49 Let us recall that 1198681198880is a quasilinear space with

the partial order relation ldquosuberdquo If we take119883 = (1198681198880)119904cup0where

0 = (0 0 ) isin 1198880 then 119903 minus dim119883 = 0 and 119904 minus dim119883 = infin

The quantity of ql-independent elements in 119883 is not finiteIndeed the family

([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)

is ql-independent Let us show that any finite subset of thisfamily is ql-independent This also implies that this family isql-independent assume that

0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)


Example 50 For the QLS 119883 = 1198681198880 119903 minus dim119883 = infin and 119904 minus

dim119883 = infin

Example 51 Wecan say that 119903minusdim119883 = 2 and 119904minusdim119883 = infinfor the QLS

119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1

0 0 ) The sequence (119864119899) is a Schauder basis for 119868119888

0

and 119868119897119901for (119901 le 1)

Proof Now we show that (119864119899) is a Schauder basis for 119868119888

0 the

other proof is similar to 1198681198880 Let 119884 = (119884

119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883

119899 From here we get

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)

we have ℎ(119885sum119899119896=1120572119896119864119896) rarr 0 for (119899 rarr infin) Now we show

that this representation is uniqueTo prove uniqueness we assume that

119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)

for 120572119896= 120573119896 1 le 119896 lt infin Since 119885 isin (119868119888

0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817

=1003817100381710038171003817(1205721 minus 1205731 1205722 minus 1205732 120572119899 minus 120573119899 )

1003817100381710038171003817

= sup1le119896ltinfin

1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573

119896for every 1 le 119896 lt infin This proves the

theorem

Example 53 (119864119899) sequence given in Theorem 52 is an

orthonormal basis for 1198681198880

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

References

[1] S M Aseev ldquoQuasilinear operators and their application inthe theory of multivalued mappingsrdquo Trudy MatematicheskogoInstituta imeni VA Steklova vol 167 pp 25ndash52 1985

[2] G Alefeld and G Mayer ldquoInterval analysis theory and applica-tionsrdquo Journal of Computational and Applied Mathematics vol121 no 1-2 pp 421ndash464 2000

[3] V LakshmikanthamTGnanaBhaskar and J VasundharaDeviTheory of Set Differential Equations inMetric Spaces CambridgeScientific Publishers Cambridge UK 2006

[4] H Bozkurt S Cakan andYYılmaz ldquoQuasilinear inner productspaces and Hilbert quasilinear spacesrdquo International Journal ofAnalysis vol 2014 Article ID 258389 2014

[5] Y Yılmaz S Cakan and S Aytekin ldquoTopological quasilinearspacesrdquo Abstract and Applied Analysis vol 2012 Article ID951374 10 pages 2012

[6] S Cakan and Y Yılmaz ldquoNormed proper quasilinear spacesrdquoJournal of Nonlinear Sciences and Applications vol 8 pp 816ndash836 2015

[7] H K Banazılı On quasilinear operators between quasilinearspaces [MS thesis] Malatya Turkey 2014

[8] R E Moore R B Kearfott and M J Cloud Introduction toInterval Analysis Society for Industrial and AppliedMathemat-ics Philadelphia Pa USA 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(6) there exists an element 120579 isin 119883 such that 119909 + 120579 = 119909(7) 120572 sdot (120573 sdot 119909) = (120572 sdot 120573) sdot 119909(8) 120572 sdot (119909 + 119910) = 120572 sdot 119909 + 120572 sdot 119910(9) 1 sdot 119909 = 119909(10) 0 sdot 119909 = 120579(11) (120572 + 120573) sdot 119909 le 120572 sdot 119909 + 120573 sdot 119909(12) 119909 + 119911 le 119910 + V if 119909 le 119910 and 119911 le V(13) 120572 sdot 119909 le 120572 sdot 119910 if 119909 le 119910

A linear space is a quasilinear space with the partial orderrelation ldquo=rdquo The most popular example which is not a linearspace is the set of all closed intervals of real numbers with theinclusion relation ldquosuberdquo algebraic sum operation

119860 + 119861 = 119886 + 119887 119886 isin 119860 119887 isin 119861 (1)

and the real-scalar multiplication

120582119860 = 120582119886 119886 isin 119860 (2)

We denote this set byΩ119862(R) Another one isΩ(R) the set of

all compact subsets of real numbers By a slight modificationof algebraic sum operation (with closure) such as

119860 + 119861 = 119886 + 119887 119886 isin 119860 119887 isin 119861 (3)

and by the same real-scalar multiplication defined above andby the inclusion relation we get the nonlinear QLS Ω

119862(119864)

and Ω(119864) the space of all nonempty closed bounded andconvex closed bounded subsets of some normed linear space119864 respectively

Lemma 2 Suppose that any element 119909 in a QLS 119883 has aninverse element 1199091015840 isin 119883 Then the partial order in 119883 is deter-mined by equality the distributivity conditions hold andconsequently119883 is a linear space [1]

Suppose that 119883 is a QLS and 119884 sube 119883 Then 119884 is calleda subspace of 119883 whenever 119884 is a QLS with the same partialorder and the restriction to 119884 of the operations on 119883 Onecan easily prove the following theorem using the condition ofbeing a QLS It is quite similar to its linear space analogue

Theorem 3 119884 is a subspace of a QLS119883 if and only if for every119909 119910 isin 119884 and 120572 120573 isin 119877 120572119909 + 120573119910 isin 119884 [5]

Let 119883 be a QLS An 119909 isin 119883 is said to be symmetric ifminus119909 = 119909 where minus119909 = (minus1)119909 and 119883

119889denotes the set of all

such elements 120579 denotes the zerorsquos additive unit of119883 and it isminimal that is 119909 = 120579 if 119909 le 120579 An element 1199091015840 is called inverseof 119909 if 119909 + 1199091015840 = 120579 The inverse is unique whenever it existsand 1199091015840 = minus119909 in this case Sometimes 1199091015840 may not exist but minus119909is always meaningful in QLSs An element 119909 possessing aninverse is called regular otherwise it is called singular For asingular element 119909 we should note that 119909 minus 119909 = 0 Now 119883

119903

and 119883119904stand for the sets of all regular and singular elements

in119883 respectively Further119883119903 119883

119889 and119883

119904cup0 are subspaces

of 119883 and they are called regular symmetric and singular

subspaces of119883 respectively [5] It is easy from the definitionsthat119883

119889sub 119883

119904and119883 = 119883

119903cup 119883

119904

In a linearQLS that is in a linear space there is no singularelement Further in a QLS119883 it is obvious that any element 119909is regular if and only if 119909 minus 119909 = 0

Proposition 4 In a QLS119883 if 119909 isin 119883 and 119910 isin 119883119904then 119909+119910 isin

119883119904

Proposition 5 In a quasilinear space119883 every regular elementis minimal [5]

Definition 6 Let119883 be a QLS and 119910 isin 119883 The set of all regularelements preceding 119910 is called floor of 119910 and 119865

119910denotes the

set of all such elements Therefore

119865119910= 119911 isin 119883

119903 119911 le 119910 (4)

The floor of any subset 119872 of 119883 is the union of floors of allelements in119872 and is denoted by 119865

119872[6]

Definition 7 Let 119883 be a QLS 119872 sube 119883 and 119909 119910 isin 119872 119872 iscalled a proper set if the following two conditions hold

(i) 119865119911= 0 for all 119911 isin 119872

(ii) 119865119909= 119865119910for each pair of points 119909 119910 with 119909 = 119910

Otherwise119872 is called an improper set Particularly if 119883is a proper set then it is called a proper quasilinear space(briefly proper qls) [6]

Definition 8 Let119883 be a quasilinear space119883 is called a solid-floored quasilinear space whenever

119910 = sup 119909 isin 119883119903 119909 le 119910 (5)

for each 119910 isin 119883 Otherwise 119883 is called a non-solid-flooredquasilinear space [6]

Definition 9 Let 119883 be a QLS Consolidation of floor of 119883 isthe smallest solid-flooredQLS containing119883 that is if thereexists another solid-floored QLS 119884 containing119883 then sube 119884

Clearly = 119883 for some solid-floored QLS 119883 FurtherΩ119862(R119899

)119904= Ω

119862(R119899

) For a QLS119883 the set

119865

119910= 119911 isin ()

119903

119911 ⪯ 119910 (6)

is the floor of119883 in

Definition 10 Let119883 be a QLS A real function sdot119883 119883 rarr 119864

1

is called a norm if the following conditions hold [1]

(14) 119909119883gt 0 if 119909 = 0

(15) 119909 + 119910119883le 119909

119883+ 119910

119883

(16) 120572119909119883= |120572|119909

119883

(17) if 119909 le 119910 then 119909119883le 119910

119883

(18) if for any 120576 gt 0 there exists an element 119909120576isin 119883 such

that 119909 le 119910 + 119909120576and 119909

120576119883le 120576 then 119909 le 119910




ℎ119883(119909 119910)

= inf 119903 ge 0 119909 le 119910 + 1198861199031 119910 le 119909 + 119886

119903

21003817100381710038171003817119886

119903

119894

1003817100381710038171003817 le 119903

(7)


ℎ119883(119909 119910) le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119883 (8)


the metric axioms



119860Ω(119864)= sup

119886isin119860

119886119864 (9)



ℎΩ119862(119864)

(119860 119861)

= inf 119903 ge 0 119860 sub 119861 + 119878119903(120579) 119861 sub 119860 + 119878

119903(120579)

(10)





119862(R)

119903equiv R


119903and ⟨119909 119909⟩ = 0 hArr 119909 = 0

(24) ⟨119909 119910⟩Ω(R) = sup⟨119886 119887⟩Ω(R) 119886 isin 119865

119909 119887 isin 119865

119910


120576isin 119883 such

that 119909 ⪯ 119910 + 119909120576and ⟨119909

120576 119909

120576⟩ sube 119878

120576(120579) then 119909 ⪯ 119910




⟨119860 119861⟩ = 119886 sdot 119887 119886 isin 119860 119887 isin 119861 (11)


119909 = radic⟨119909 119909⟩Ω(R) (12)

for every 119909 isin 119883 [4]


119899rarr 119910 then

⟨119909119899 119910

119899⟩ rarr ⟨119909 119910⟩ [4]




1003817100381710038171003817⟨119909 119910⟩1003817100381710038171003817Ω(R)

= 0 (13)



1003817100381710038171003817⟨119909 119910⟩1003817100381710038171003817Ω(R)

=

0 119909 = 119910

1 119909 = 119910

(14)



119860perp

= 119909 isin 119883 1003817100381710038171003817⟨119909 119910⟩

1003817100381710038171003817Ω(R)= 0 119910 isin 119860 (15)



119896=1sub 119883 The


12057211199091+ 120572

21199092+ sdot sdot sdot + 120572

119899119909119899le 119909 (16)


119896119899


119896119899


of 119909119896119899


119896119899

119896=1may not be





is unique The set

Qsp119860 = 119909 isin 119883 119899

sum

119896=1

120572119896119909119896le 119909 119909

1 119909

2 119909

119899

isin 119860 1205721 120572

2 120572

119899isin R

(17)




119896=1sub 119883 and 120582

119896119899

119896=1sub R

If

120579 le 12058211199091+ 120582

21199092+ sdot sdot sdot + 120582

119899119909119899

(18)

implies 1205821= 120582


119899= 0 then 119909

119896119899

119896=1is said to



119862(R119899

) [7]








119911 =

infin

sum

119899=1

120572(119911)

119899119909119899

119910 = sup 119909 isin 119883119903 119909 le 119910

(19)



ℎ(119911

119896

sum

119899=1

120572(119911)

119899119909119899) 997888rarr 0 (119896 997888rarr infin)

119910 = sup 119909 isin 119883119903 119909 le 119910

(20)




quasilinear space



3 Interval Spaces


= (1198831 119883

2) 119883

1 119883


can give

119868R119899

= (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin 119868R (21)


Example 27 Let119883 = (1198831 119883

2 119883

119899) isin 119868R119899 and119884 = (119884

1 119884

2


by the expression

119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899) (22)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899) (23)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (24)



119862(R119899



119862(R2

) So 119868R119899 and Ω119862(R119899

)



119894isin R for every 119861 = (119861

1 119861

2 119861

119899) isin


119894= 119887119894for 1 le 119894 le 119899 and 119886

119894 119887119894isin R then 119861 = ([119886

1 1198871]

[1198862 1198872] [119886



119883 =1003817100381710038171003817(1198831 1198832 119883119899)

1003817100381710038171003817 = sup1le119894le119899

10038171003817100381710038171198831198941003817100381710038171003817119868R (25)



119909isin119883|119909|



ℎ119868R (119860 119861)

= inf 119903 ge 0 119860 sube 119861 + 119878119903(120579) 119861 sube 119860 + 119878

119903(120579)

(26)



1le119894le119899119883

119894119868R


119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(27)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(28)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883

119899) isin 119868R119899 From (28) by


119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598 119895 = 1 2

(29)


119894119868R

Also we obtain

119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(30)



(


119883 = sup 100381710038171003817100381711988311003817100381710038171003817 100381710038171003817100381711988321003817100381710038171003817

10038171003817100381710038171198831198991003817100381710038171003817 le10038171003817100381710038171198611198831003817100381710038171003817 = 1 (31)

we have 1198831 le 1 119883

2 le 1 119883

119899 le 1 hArr 119883

1sube [minus1 1]

1198832sube [minus1 1] 119883



1198832= (

119899

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(32)


⟨119883 119884⟩ =

119899

sum

119894=1

⟨119883119894 119884

119894⟩119868R (33)



norm


(sum119899

119894=1119883

1198942



119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(34)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(35)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

=120598

radic119899 119895 = 1 2

(36)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(37)


119899


Example 33 Let 119883 = (119868R2


Qsp119860 = 119883 isin 119868R2





1 120582

2isin R This

means Qsp119861 = 119883




1sdot (1 0) + 120572

2sdot (0 1)



(sum2

119894=1119883

1198942





Floor of 119860 is

119865119860= (119910

1 119910

2) isin (119868R

2

)119903

(1199101 119910

2) sube 119860

= (1199101 119910

2) isin R

2

minus1 le 1199101le 1 0 le 119910

2le 2

(39)

Floor of119883 = (1198831 119883

2 119883

119899) isin 119868R119899 is

119865119868R119899 = ⋃119865119883 = ⋃(1199101 1199102 119910119899) isin (119868R

119899

)119903 119910

1

isin 1198831 119910

2isin 119883

2 119910

119899isin 119883

119899

(40)



2 119883

119899) and


an element we have

119865119883= 119884 isin (119868R

119899

)119903 119884 le 119883 (41)



119894isin 119868R for every 119883 =

(1198831 119883

2 119883


119884119894isin R


119883119894= sup119865

119883119894= sup 119884

119894isin R 119884

119894le 119883

119894 (42)


119883for every119883 = (119883

1 119883

2 119883

119899) isin 119868R119899

Example 39 The set 119882 = 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2

119883119899isin (119868R)



119881

= 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119900119889

(43)




(1198601 119860

2 119860

119899) 119861 = (119861

1 119861

2 119861

119899) isin 119868R119899 such that

119860 = 119861 If 119860 ⊊ 119861 then 119860119894⊊ 119861



119894such that 119886

119894notin 119861

119894 So 119886

119894 sube 119860

119894and

119886119894 ⊊ 119861


(1198861 119886

2 119886

119899) sube (119860

1 119860

2 119860

119899)

(1198861 119886

2 119886

119899) ⊊ (119861

1 119861

2 119861

119899)

(44)

If 119861 ⊊ 119860 then 119861119894⊊ 119860


least 119887119894isin 119861

119894such that 119887

119894notin 119860

119894 So 119887

119894 sube 119861

119894and 119887

119894 ⊊ 119860

119894 From

here we have (1198871 119887

2 119887

119899) sube (119861

1 119861

2 119861

119899) = 119861

and (1198871 119887

2 119887

119899) ⊊ (119860

1 119860

2 119860

119899) = 119860 If there


119894and 119861



119894 119886

119894notin 119861

119894and 119887

119894isin 119861

119894

119887119894notin 119860


119894 sube 119860

119894 119886

119894 ⊊

119861119894and 119887

119894 sube 119861

119894 119887

119894 ⊊ 119860


have (1198861 119886

2 119886

119899) sube 119860 (119886

1 119886

2 119886

119899) ⊊ 119861 and

(1198871 119887

2 119887

119899) sube 119861 (119887

1 119887

2 119887

119899) ⊊ 119860 So 119868R119899 is



(119868R119899

)119904

= 0

cup (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119904

(45)



119868R119899

(119868R119899

)119904 and (119868R119899


119903 minus dim (R119899

) = 119899

119904 minus dim (R119899

) = 0

119903 minus dim (119868R119899

) = 119899

119904 minus dim (119868R119899

) = 119899

119903 minus dim (119868R119899

)119904= 0

119904 minus dim (119868R119899

)119904= 119899


119903 minus dim (119868R119899

)119903= 119899

119904 minus dim (119868R119899

)119903= 0

(46)


)119904cup


and 1198862= (0 [2 4]) of 119860

119904 The set 119886



1and 120582

2satisfying the

inclusion (0 0) sube 12058211198861+ 120582





119868119904 = 119883 = (119883119899) isin 119868R

infin

(119883119899) = (119883

1 119883

2 119883

119899 )

isin 119868Rinfin

(47)


2

119883119899) isin 119868R119899 and 119884 = (119884

1 119884

2 119884



119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899 ) (48)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899 ) (49)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (50)


119889 (119883 119884) =

infin

sum

119894=1

1

2119894[ℎ (119883

119894 119884

119894)

1 + ℎ (119883119894 119884

119894)] (51)



119868119897infin= 119883 = (119883

119899) isin 119868R

infin

(119883119899) bounded (52)




infin (53)




sup 1003817100381710038171003817119883119894 + 1198841198941003817100381710038171003817119868R le sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R+10038171003817100381710038171198841198941003817100381710038171003817119868R

le sup 10038171003817100381710038171198831198941003817100381710038171003817119868R + sup 1003817100381710038171003817119884119894

1003817100381710038171003817119868R

le infin

sup 10038171003817100381710038171205821198831198941003817100381710038171003817119868R le sup 120582 1003817100381710038171003817119883119894

1003817100381710038171003817119868R le 120582 sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R

le infin

(54)

119868119897infin


119883 = sup 10038171003817100381710038171198831198991003817100381710038171003817119868R 119899 isin N (55)


120598and 119883

120598 le 120598 for every




119894+ 119883



119883119894le 119884


Denote

1198681198880= 119883 = (119883

119899) isin 119868R

infin

(119883119899) 997888rarr 0 (56)


119899)


lim119899rarrinfin

10038171003817100381710038171198831198991003817100381710038171003817119868R= 0 lArrrArr 119883 isin 119868119888

0 (57)



(50)The space 119868119888

0with the norm

119883 = sup119894isinN

10038171003817100381710038171198831198941003817100381710038171003817119868R (58)


defined by

1198681198972= 119883 = (119883

119899) isin 119868R

infin

infin

sum

119894=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868Rbounded (59)



119889 (119883 119884)

= inf 119903 ge 0 119883 sube 119884 + 119878119903(120579) 119884 sube 119883 + 119878

119903(120579)

(60)

where 119878119903(120579) = 119883 isin 119868119897

2 119883 le 119903


119883 = (

infin

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(61)


ear space with (61)






(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)




2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)


119889 (119883119899

119894 119883

119898

119894) le 120598 (64)


119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)


0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)



119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)


119894=1119883

1198942

le infin



119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897




2hArr sum

infin

119899=1119883

1198992


119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)


119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)


119883119894


119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)


since 1198681198972is a




0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904





([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)


0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)



dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ℎ119883(119909 119910)

= inf 119903 ge 0 119909 le 119910 + 1198861199031 119910 le 119909 + 119886

119903

21003817100381710038171003817119886

119903

119894

1003817100381710038171003817 le 119903

(7)


ℎ119883(119909 119910) le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817119883 (8)


the metric axioms



119860Ω(119864)= sup

119886isin119860

119886119864 (9)



ℎΩ119862(119864)

(119860 119861)

= inf 119903 ge 0 119860 sub 119861 + 119878119903(120579) 119861 sub 119860 + 119878

119903(120579)

(10)





119862(R)

119903equiv R


119903and ⟨119909 119909⟩ = 0 hArr 119909 = 0

(24) ⟨119909 119910⟩Ω(R) = sup⟨119886 119887⟩Ω(R) 119886 isin 119865

119909 119887 isin 119865

119910


120576isin 119883 such

that 119909 ⪯ 119910 + 119909120576and ⟨119909

120576 119909

120576⟩ sube 119878

120576(120579) then 119909 ⪯ 119910




⟨119860 119861⟩ = 119886 sdot 119887 119886 isin 119860 119887 isin 119861 (11)


119909 = radic⟨119909 119909⟩Ω(R) (12)

for every 119909 isin 119883 [4]


119899rarr 119910 then

⟨119909119899 119910

119899⟩ rarr ⟨119909 119910⟩ [4]




1003817100381710038171003817⟨119909 119910⟩1003817100381710038171003817Ω(R)

= 0 (13)



1003817100381710038171003817⟨119909 119910⟩1003817100381710038171003817Ω(R)

=

0 119909 = 119910

1 119909 = 119910

(14)



119860perp

= 119909 isin 119883 1003817100381710038171003817⟨119909 119910⟩

1003817100381710038171003817Ω(R)= 0 119910 isin 119860 (15)



119896=1sub 119883 The


12057211199091+ 120572

21199092+ sdot sdot sdot + 120572

119899119909119899le 119909 (16)


119896119899


119896119899


of 119909119896119899


119896119899

119896=1may not be





is unique The set

Qsp119860 = 119909 isin 119883 119899

sum

119896=1

120572119896119909119896le 119909 119909

1 119909

2 119909

119899

isin 119860 1205721 120572

2 120572

119899isin R

(17)




119896=1sub 119883 and 120582

119896119899

119896=1sub R

If

120579 le 12058211199091+ 120582

21199092+ sdot sdot sdot + 120582

119899119909119899

(18)

implies 1205821= 120582


119899= 0 then 119909

119896119899

119896=1is said to



119862(R119899

) [7]








119911 =

infin

sum

119899=1

120572(119911)

119899119909119899

119910 = sup 119909 isin 119883119903 119909 le 119910

(19)



ℎ(119911

119896

sum

119899=1

120572(119911)

119899119909119899) 997888rarr 0 (119896 997888rarr infin)

119910 = sup 119909 isin 119883119903 119909 le 119910

(20)




quasilinear space



3 Interval Spaces


= (1198831 119883

2) 119883

1 119883


can give

119868R119899

= (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin 119868R (21)


Example 27 Let119883 = (1198831 119883

2 119883

119899) isin 119868R119899 and119884 = (119884

1 119884

2


by the expression

119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899) (22)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899) (23)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (24)



119862(R119899



119862(R2

) So 119868R119899 and Ω119862(R119899

)



119894isin R for every 119861 = (119861

1 119861

2 119861

119899) isin


119894= 119887119894for 1 le 119894 le 119899 and 119886

119894 119887119894isin R then 119861 = ([119886

1 1198871]

[1198862 1198872] [119886



119883 =1003817100381710038171003817(1198831 1198832 119883119899)

1003817100381710038171003817 = sup1le119894le119899

10038171003817100381710038171198831198941003817100381710038171003817119868R (25)



119909isin119883|119909|



ℎ119868R (119860 119861)

= inf 119903 ge 0 119860 sube 119861 + 119878119903(120579) 119861 sube 119860 + 119878

119903(120579)

(26)



1le119894le119899119883

119894119868R


119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(27)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(28)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883

119899) isin 119868R119899 From (28) by


119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598 119895 = 1 2

(29)


119894119868R

Also we obtain

119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(30)



(


119883 = sup 100381710038171003817100381711988311003817100381710038171003817 100381710038171003817100381711988321003817100381710038171003817

10038171003817100381710038171198831198991003817100381710038171003817 le10038171003817100381710038171198611198831003817100381710038171003817 = 1 (31)

we have 1198831 le 1 119883

2 le 1 119883

119899 le 1 hArr 119883

1sube [minus1 1]

1198832sube [minus1 1] 119883



1198832= (

119899

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(32)


⟨119883 119884⟩ =

119899

sum

119894=1

⟨119883119894 119884

119894⟩119868R (33)



norm


(sum119899

119894=1119883

1198942



119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(34)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(35)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

=120598

radic119899 119895 = 1 2

(36)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(37)


119899


Example 33 Let 119883 = (119868R2


Qsp119860 = 119883 isin 119868R2





1 120582

2isin R This

means Qsp119861 = 119883




1sdot (1 0) + 120572

2sdot (0 1)



(sum2

119894=1119883

1198942





Floor of 119860 is

119865119860= (119910

1 119910

2) isin (119868R

2

)119903

(1199101 119910

2) sube 119860

= (1199101 119910

2) isin R

2

minus1 le 1199101le 1 0 le 119910

2le 2

(39)

Floor of119883 = (1198831 119883

2 119883

119899) isin 119868R119899 is

119865119868R119899 = ⋃119865119883 = ⋃(1199101 1199102 119910119899) isin (119868R

119899

)119903 119910

1

isin 1198831 119910

2isin 119883

2 119910

119899isin 119883

119899

(40)



2 119883

119899) and


an element we have

119865119883= 119884 isin (119868R

119899

)119903 119884 le 119883 (41)



119894isin 119868R for every 119883 =

(1198831 119883

2 119883


119884119894isin R


119883119894= sup119865

119883119894= sup 119884

119894isin R 119884

119894le 119883

119894 (42)


119883for every119883 = (119883

1 119883

2 119883

119899) isin 119868R119899

Example 39 The set 119882 = 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2

119883119899isin (119868R)



119881

= 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119900119889

(43)




(1198601 119860

2 119860

119899) 119861 = (119861

1 119861

2 119861

119899) isin 119868R119899 such that

119860 = 119861 If 119860 ⊊ 119861 then 119860119894⊊ 119861



119894such that 119886

119894notin 119861

119894 So 119886

119894 sube 119860

119894and

119886119894 ⊊ 119861


(1198861 119886

2 119886

119899) sube (119860

1 119860

2 119860

119899)

(1198861 119886

2 119886

119899) ⊊ (119861

1 119861

2 119861

119899)

(44)

If 119861 ⊊ 119860 then 119861119894⊊ 119860


least 119887119894isin 119861

119894such that 119887

119894notin 119860

119894 So 119887

119894 sube 119861

119894and 119887

119894 ⊊ 119860

119894 From

here we have (1198871 119887

2 119887

119899) sube (119861

1 119861

2 119861

119899) = 119861

and (1198871 119887

2 119887

119899) ⊊ (119860

1 119860

2 119860

119899) = 119860 If there


119894and 119861



119894 119886

119894notin 119861

119894and 119887

119894isin 119861

119894

119887119894notin 119860


119894 sube 119860

119894 119886

119894 ⊊

119861119894and 119887

119894 sube 119861

119894 119887

119894 ⊊ 119860


have (1198861 119886

2 119886

119899) sube 119860 (119886

1 119886

2 119886

119899) ⊊ 119861 and

(1198871 119887

2 119887

119899) sube 119861 (119887

1 119887

2 119887

119899) ⊊ 119860 So 119868R119899 is



(119868R119899

)119904

= 0

cup (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119904

(45)



119868R119899

(119868R119899

)119904 and (119868R119899


119903 minus dim (R119899

) = 119899

119904 minus dim (R119899

) = 0

119903 minus dim (119868R119899

) = 119899

119904 minus dim (119868R119899

) = 119899

119903 minus dim (119868R119899

)119904= 0

119904 minus dim (119868R119899

)119904= 119899


119903 minus dim (119868R119899

)119903= 119899

119904 minus dim (119868R119899

)119903= 0

(46)


)119904cup


and 1198862= (0 [2 4]) of 119860

119904 The set 119886



1and 120582

2satisfying the

inclusion (0 0) sube 12058211198861+ 120582





119868119904 = 119883 = (119883119899) isin 119868R

infin

(119883119899) = (119883

1 119883

2 119883

119899 )

isin 119868Rinfin

(47)


2

119883119899) isin 119868R119899 and 119884 = (119884

1 119884

2 119884



119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899 ) (48)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899 ) (49)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (50)


119889 (119883 119884) =

infin

sum

119894=1

1

2119894[ℎ (119883

119894 119884

119894)

1 + ℎ (119883119894 119884

119894)] (51)



119868119897infin= 119883 = (119883

119899) isin 119868R

infin

(119883119899) bounded (52)




infin (53)




sup 1003817100381710038171003817119883119894 + 1198841198941003817100381710038171003817119868R le sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R+10038171003817100381710038171198841198941003817100381710038171003817119868R

le sup 10038171003817100381710038171198831198941003817100381710038171003817119868R + sup 1003817100381710038171003817119884119894

1003817100381710038171003817119868R

le infin

sup 10038171003817100381710038171205821198831198941003817100381710038171003817119868R le sup 120582 1003817100381710038171003817119883119894

1003817100381710038171003817119868R le 120582 sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R

le infin

(54)

119868119897infin


119883 = sup 10038171003817100381710038171198831198991003817100381710038171003817119868R 119899 isin N (55)


120598and 119883

120598 le 120598 for every




119894+ 119883



119883119894le 119884


Denote

1198681198880= 119883 = (119883

119899) isin 119868R

infin

(119883119899) 997888rarr 0 (56)


119899)


lim119899rarrinfin

10038171003817100381710038171198831198991003817100381710038171003817119868R= 0 lArrrArr 119883 isin 119868119888

0 (57)



(50)The space 119868119888

0with the norm

119883 = sup119894isinN

10038171003817100381710038171198831198941003817100381710038171003817119868R (58)


defined by

1198681198972= 119883 = (119883

119899) isin 119868R

infin

infin

sum

119894=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868Rbounded (59)



119889 (119883 119884)

= inf 119903 ge 0 119883 sube 119884 + 119878119903(120579) 119884 sube 119883 + 119878

119903(120579)

(60)

where 119878119903(120579) = 119883 isin 119868119897

2 119883 le 119903


119883 = (

infin

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(61)


ear space with (61)






(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)




2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)


119889 (119883119899

119894 119883

119898

119894) le 120598 (64)


119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)


0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)



119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)


119894=1119883

1198942

le infin



119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897




2hArr sum

infin

119899=1119883

1198992


119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)


119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)


119883119894


119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)


since 1198681198972is a




0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904





([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)


0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)



dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












is unique The set

Qsp119860 = 119909 isin 119883 119899

sum

119896=1

120572119896119909119896le 119909 119909

1 119909

2 119909

119899

isin 119860 1205721 120572

2 120572

119899isin R

(17)




119896=1sub 119883 and 120582

119896119899

119896=1sub R

If

120579 le 12058211199091+ 120582

21199092+ sdot sdot sdot + 120582

119899119909119899

(18)

implies 1205821= 120582


119899= 0 then 119909

119896119899

119896=1is said to



119862(R119899

) [7]








119911 =

infin

sum

119899=1

120572(119911)

119899119909119899

119910 = sup 119909 isin 119883119903 119909 le 119910

(19)



ℎ(119911

119896

sum

119899=1

120572(119911)

119899119909119899) 997888rarr 0 (119896 997888rarr infin)

119910 = sup 119909 isin 119883119903 119909 le 119910

(20)




quasilinear space



3 Interval Spaces


= (1198831 119883

2) 119883

1 119883


can give

119868R119899

= (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin 119868R (21)


Example 27 Let119883 = (1198831 119883

2 119883

119899) isin 119868R119899 and119884 = (119884

1 119884

2


by the expression

119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899) (22)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899) (23)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (24)



119862(R119899



119862(R2

) So 119868R119899 and Ω119862(R119899

)



119894isin R for every 119861 = (119861

1 119861

2 119861

119899) isin


119894= 119887119894for 1 le 119894 le 119899 and 119886

119894 119887119894isin R then 119861 = ([119886

1 1198871]

[1198862 1198872] [119886



119883 =1003817100381710038171003817(1198831 1198832 119883119899)

1003817100381710038171003817 = sup1le119894le119899

10038171003817100381710038171198831198941003817100381710038171003817119868R (25)



119909isin119883|119909|



ℎ119868R (119860 119861)

= inf 119903 ge 0 119860 sube 119861 + 119878119903(120579) 119861 sube 119860 + 119878

119903(120579)

(26)



1le119894le119899119883

119894119868R


119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(27)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(28)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883

119899) isin 119868R119899 From (28) by


119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598 119895 = 1 2

(29)


119894119868R

Also we obtain

119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(30)



(


119883 = sup 100381710038171003817100381711988311003817100381710038171003817 100381710038171003817100381711988321003817100381710038171003817

10038171003817100381710038171198831198991003817100381710038171003817 le10038171003817100381710038171198611198831003817100381710038171003817 = 1 (31)

we have 1198831 le 1 119883

2 le 1 119883

119899 le 1 hArr 119883

1sube [minus1 1]

1198832sube [minus1 1] 119883



1198832= (

119899

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(32)


⟨119883 119884⟩ =

119899

sum

119894=1

⟨119883119894 119884

119894⟩119868R (33)



norm


(sum119899

119894=1119883

1198942



119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(34)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(35)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

=120598

radic119899 119895 = 1 2

(36)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(37)


119899


Example 33 Let 119883 = (119868R2


Qsp119860 = 119883 isin 119868R2





1 120582

2isin R This

means Qsp119861 = 119883




1sdot (1 0) + 120572

2sdot (0 1)



(sum2

119894=1119883

1198942





Floor of 119860 is

119865119860= (119910

1 119910

2) isin (119868R

2

)119903

(1199101 119910

2) sube 119860

= (1199101 119910

2) isin R

2

minus1 le 1199101le 1 0 le 119910

2le 2

(39)

Floor of119883 = (1198831 119883

2 119883

119899) isin 119868R119899 is

119865119868R119899 = ⋃119865119883 = ⋃(1199101 1199102 119910119899) isin (119868R

119899

)119903 119910

1

isin 1198831 119910

2isin 119883

2 119910

119899isin 119883

119899

(40)



2 119883

119899) and


an element we have

119865119883= 119884 isin (119868R

119899

)119903 119884 le 119883 (41)



119894isin 119868R for every 119883 =

(1198831 119883

2 119883


119884119894isin R


119883119894= sup119865

119883119894= sup 119884

119894isin R 119884

119894le 119883

119894 (42)


119883for every119883 = (119883

1 119883

2 119883

119899) isin 119868R119899

Example 39 The set 119882 = 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2

119883119899isin (119868R)



119881

= 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119900119889

(43)




(1198601 119860

2 119860

119899) 119861 = (119861

1 119861

2 119861

119899) isin 119868R119899 such that

119860 = 119861 If 119860 ⊊ 119861 then 119860119894⊊ 119861



119894such that 119886

119894notin 119861

119894 So 119886

119894 sube 119860

119894and

119886119894 ⊊ 119861


(1198861 119886

2 119886

119899) sube (119860

1 119860

2 119860

119899)

(1198861 119886

2 119886

119899) ⊊ (119861

1 119861

2 119861

119899)

(44)

If 119861 ⊊ 119860 then 119861119894⊊ 119860


least 119887119894isin 119861

119894such that 119887

119894notin 119860

119894 So 119887

119894 sube 119861

119894and 119887

119894 ⊊ 119860

119894 From

here we have (1198871 119887

2 119887

119899) sube (119861

1 119861

2 119861

119899) = 119861

and (1198871 119887

2 119887

119899) ⊊ (119860

1 119860

2 119860

119899) = 119860 If there


119894and 119861



119894 119886

119894notin 119861

119894and 119887

119894isin 119861

119894

119887119894notin 119860


119894 sube 119860

119894 119886

119894 ⊊

119861119894and 119887

119894 sube 119861

119894 119887

119894 ⊊ 119860


have (1198861 119886

2 119886

119899) sube 119860 (119886

1 119886

2 119886

119899) ⊊ 119861 and

(1198871 119887

2 119887

119899) sube 119861 (119887

1 119887

2 119887

119899) ⊊ 119860 So 119868R119899 is



(119868R119899

)119904

= 0

cup (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119904

(45)



119868R119899

(119868R119899

)119904 and (119868R119899


119903 minus dim (R119899

) = 119899

119904 minus dim (R119899

) = 0

119903 minus dim (119868R119899

) = 119899

119904 minus dim (119868R119899

) = 119899

119903 minus dim (119868R119899

)119904= 0

119904 minus dim (119868R119899

)119904= 119899


119903 minus dim (119868R119899

)119903= 119899

119904 minus dim (119868R119899

)119903= 0

(46)


)119904cup


and 1198862= (0 [2 4]) of 119860

119904 The set 119886



1and 120582

2satisfying the

inclusion (0 0) sube 12058211198861+ 120582





119868119904 = 119883 = (119883119899) isin 119868R

infin

(119883119899) = (119883

1 119883

2 119883

119899 )

isin 119868Rinfin

(47)


2

119883119899) isin 119868R119899 and 119884 = (119884

1 119884

2 119884



119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899 ) (48)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899 ) (49)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (50)


119889 (119883 119884) =

infin

sum

119894=1

1

2119894[ℎ (119883

119894 119884

119894)

1 + ℎ (119883119894 119884

119894)] (51)



119868119897infin= 119883 = (119883

119899) isin 119868R

infin

(119883119899) bounded (52)




infin (53)




sup 1003817100381710038171003817119883119894 + 1198841198941003817100381710038171003817119868R le sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R+10038171003817100381710038171198841198941003817100381710038171003817119868R

le sup 10038171003817100381710038171198831198941003817100381710038171003817119868R + sup 1003817100381710038171003817119884119894

1003817100381710038171003817119868R

le infin

sup 10038171003817100381710038171205821198831198941003817100381710038171003817119868R le sup 120582 1003817100381710038171003817119883119894

1003817100381710038171003817119868R le 120582 sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R

le infin

(54)

119868119897infin


119883 = sup 10038171003817100381710038171198831198991003817100381710038171003817119868R 119899 isin N (55)


120598and 119883

120598 le 120598 for every




119894+ 119883



119883119894le 119884


Denote

1198681198880= 119883 = (119883

119899) isin 119868R

infin

(119883119899) 997888rarr 0 (56)


119899)


lim119899rarrinfin

10038171003817100381710038171198831198991003817100381710038171003817119868R= 0 lArrrArr 119883 isin 119868119888

0 (57)



(50)The space 119868119888

0with the norm

119883 = sup119894isinN

10038171003817100381710038171198831198941003817100381710038171003817119868R (58)


defined by

1198681198972= 119883 = (119883

119899) isin 119868R

infin

infin

sum

119894=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868Rbounded (59)



119889 (119883 119884)

= inf 119903 ge 0 119883 sube 119884 + 119878119903(120579) 119884 sube 119883 + 119878

119903(120579)

(60)

where 119878119903(120579) = 119883 isin 119868119897

2 119883 le 119903


119883 = (

infin

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(61)


ear space with (61)






(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)




2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)


119889 (119883119899

119894 119883

119898

119894) le 120598 (64)


119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)


0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)



119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)


119894=1119883

1198942

le infin



119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897




2hArr sum

infin

119899=1119883

1198992


119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)


119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)


119883119894


119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)


since 1198681198972is a




0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904





([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)


0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)



dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909isin119883|119909|



ℎ119868R (119860 119861)

= inf 119903 ge 0 119860 sube 119861 + 119878119903(120579) 119861 sube 119860 + 119878

119903(120579)

(26)



1le119894le119899119883

119894119868R


119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(27)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(28)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883

119899) isin 119868R119899 From (28) by


119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598 119895 = 1 2

(29)


119894119868R

Also we obtain

119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(30)



(


119883 = sup 100381710038171003817100381711988311003817100381710038171003817 100381710038171003817100381711988321003817100381710038171003817

10038171003817100381710038171198831198991003817100381710038171003817 le10038171003817100381710038171198611198831003817100381710038171003817 = 1 (31)

we have 1198831 le 1 119883

2 le 1 119883

119899 le 1 hArr 119883

1sube [minus1 1]

1198832sube [minus1 1] 119883



1198832= (

119899

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(32)


⟨119883 119884⟩ =

119899

sum

119894=1

⟨119883119894 119884

119894⟩119868R (33)



norm


(sum119899

119894=1119883

1198942



119883119899

le 119883119898

+ 1198601120598

119883119898

le 119883119899

+ 1198602120598

1003817100381710038171003817100381711986011989512059810038171003817100381710038171003817le 120598 119895 = 1 2

(34)



119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(35)



119883119899

119894rarr 119883


119894isin 119868R for

every 119894 isin [1 119899] 119883 = (1198831 119883

2 119883



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

=120598

radic119899 119895 = 1 2

(36)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(37)


119899


Example 33 Let 119883 = (119868R2


Qsp119860 = 119883 isin 119868R2





1 120582

2isin R This

means Qsp119861 = 119883




1sdot (1 0) + 120572

2sdot (0 1)



(sum2

119894=1119883

1198942





Floor of 119860 is

119865119860= (119910

1 119910

2) isin (119868R

2

)119903

(1199101 119910

2) sube 119860

= (1199101 119910

2) isin R

2

minus1 le 1199101le 1 0 le 119910

2le 2

(39)

Floor of119883 = (1198831 119883

2 119883

119899) isin 119868R119899 is

119865119868R119899 = ⋃119865119883 = ⋃(1199101 1199102 119910119899) isin (119868R

119899

)119903 119910

1

isin 1198831 119910

2isin 119883

2 119910

119899isin 119883

119899

(40)



2 119883

119899) and


an element we have

119865119883= 119884 isin (119868R

119899

)119903 119884 le 119883 (41)



119894isin 119868R for every 119883 =

(1198831 119883

2 119883


119884119894isin R


119883119894= sup119865

119883119894= sup 119884

119894isin R 119884

119894le 119883

119894 (42)


119883for every119883 = (119883

1 119883

2 119883

119899) isin 119868R119899

Example 39 The set 119882 = 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2

119883119899isin (119868R)



119881

= 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119900119889

(43)




(1198601 119860

2 119860

119899) 119861 = (119861

1 119861

2 119861

119899) isin 119868R119899 such that

119860 = 119861 If 119860 ⊊ 119861 then 119860119894⊊ 119861



119894such that 119886

119894notin 119861

119894 So 119886

119894 sube 119860

119894and

119886119894 ⊊ 119861


(1198861 119886

2 119886

119899) sube (119860

1 119860

2 119860

119899)

(1198861 119886

2 119886

119899) ⊊ (119861

1 119861

2 119861

119899)

(44)

If 119861 ⊊ 119860 then 119861119894⊊ 119860


least 119887119894isin 119861

119894such that 119887

119894notin 119860

119894 So 119887

119894 sube 119861

119894and 119887

119894 ⊊ 119860

119894 From

here we have (1198871 119887

2 119887

119899) sube (119861

1 119861

2 119861

119899) = 119861

and (1198871 119887

2 119887

119899) ⊊ (119860

1 119860

2 119860

119899) = 119860 If there


119894and 119861



119894 119886

119894notin 119861

119894and 119887

119894isin 119861

119894

119887119894notin 119860


119894 sube 119860

119894 119886

119894 ⊊

119861119894and 119887

119894 sube 119861

119894 119887

119894 ⊊ 119860


have (1198861 119886

2 119886

119899) sube 119860 (119886

1 119886

2 119886

119899) ⊊ 119861 and

(1198871 119887

2 119887

119899) sube 119861 (119887

1 119887

2 119887

119899) ⊊ 119860 So 119868R119899 is



(119868R119899

)119904

= 0

cup (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119904

(45)



119868R119899

(119868R119899

)119904 and (119868R119899


119903 minus dim (R119899

) = 119899

119904 minus dim (R119899

) = 0

119903 minus dim (119868R119899

) = 119899

119904 minus dim (119868R119899

) = 119899

119903 minus dim (119868R119899

)119904= 0

119904 minus dim (119868R119899

)119904= 119899


119903 minus dim (119868R119899

)119903= 119899

119904 minus dim (119868R119899

)119903= 0

(46)


)119904cup


and 1198862= (0 [2 4]) of 119860

119904 The set 119886



1and 120582

2satisfying the

inclusion (0 0) sube 12058211198861+ 120582





119868119904 = 119883 = (119883119899) isin 119868R

infin

(119883119899) = (119883

1 119883

2 119883

119899 )

isin 119868Rinfin

(47)


2

119883119899) isin 119868R119899 and 119884 = (119884

1 119884

2 119884



119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899 ) (48)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899 ) (49)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (50)


119889 (119883 119884) =

infin

sum

119894=1

1

2119894[ℎ (119883

119894 119884

119894)

1 + ℎ (119883119894 119884

119894)] (51)



119868119897infin= 119883 = (119883

119899) isin 119868R

infin

(119883119899) bounded (52)




infin (53)




sup 1003817100381710038171003817119883119894 + 1198841198941003817100381710038171003817119868R le sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R+10038171003817100381710038171198841198941003817100381710038171003817119868R

le sup 10038171003817100381710038171198831198941003817100381710038171003817119868R + sup 1003817100381710038171003817119884119894

1003817100381710038171003817119868R

le infin

sup 10038171003817100381710038171205821198831198941003817100381710038171003817119868R le sup 120582 1003817100381710038171003817119883119894

1003817100381710038171003817119868R le 120582 sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R

le infin

(54)

119868119897infin


119883 = sup 10038171003817100381710038171198831198991003817100381710038171003817119868R 119899 isin N (55)


120598and 119883

120598 le 120598 for every




119894+ 119883



119883119894le 119884


Denote

1198681198880= 119883 = (119883

119899) isin 119868R

infin

(119883119899) 997888rarr 0 (56)


119899)


lim119899rarrinfin

10038171003817100381710038171198831198991003817100381710038171003817119868R= 0 lArrrArr 119883 isin 119868119888

0 (57)



(50)The space 119868119888

0with the norm

119883 = sup119894isinN

10038171003817100381710038171198831198941003817100381710038171003817119868R (58)


defined by

1198681198972= 119883 = (119883

119899) isin 119868R

infin

infin

sum

119894=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868Rbounded (59)



119889 (119883 119884)

= inf 119903 ge 0 119883 sube 119884 + 119878119903(120579) 119884 sube 119883 + 119878

119903(120579)

(60)

where 119878119903(120579) = 119883 isin 119868119897

2 119883 le 119903


119883 = (

infin

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(61)


ear space with (61)






(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)




2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)


119889 (119883119899

119894 119883

119898

119894) le 120598 (64)


119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)


0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)



119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)


119894=1119883

1198942

le infin



119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897




2hArr sum

infin

119899=1119883

1198992


119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)


119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)


119883119894


119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)


since 1198681198972is a




0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904





([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)


0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)



dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 120582

2isin R This

means Qsp119861 = 119883




1sdot (1 0) + 120572

2sdot (0 1)



(sum2

119894=1119883

1198942





Floor of 119860 is

119865119860= (119910

1 119910

2) isin (119868R

2

)119903

(1199101 119910

2) sube 119860

= (1199101 119910

2) isin R

2

minus1 le 1199101le 1 0 le 119910

2le 2

(39)

Floor of119883 = (1198831 119883

2 119883

119899) isin 119868R119899 is

119865119868R119899 = ⋃119865119883 = ⋃(1199101 1199102 119910119899) isin (119868R

119899

)119903 119910

1

isin 1198831 119910

2isin 119883

2 119910

119899isin 119883

119899

(40)



2 119883

119899) and


an element we have

119865119883= 119884 isin (119868R

119899

)119903 119884 le 119883 (41)



119894isin 119868R for every 119883 =

(1198831 119883

2 119883


119884119894isin R


119883119894= sup119865

119883119894= sup 119884

119894isin R 119884

119894le 119883

119894 (42)


119883for every119883 = (119883

1 119883

2 119883

119899) isin 119868R119899

Example 39 The set 119882 = 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2

119883119899isin (119868R)



119881

= 119883 = (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119900119889

(43)




(1198601 119860

2 119860

119899) 119861 = (119861

1 119861

2 119861

119899) isin 119868R119899 such that

119860 = 119861 If 119860 ⊊ 119861 then 119860119894⊊ 119861



119894such that 119886

119894notin 119861

119894 So 119886

119894 sube 119860

119894and

119886119894 ⊊ 119861


(1198861 119886

2 119886

119899) sube (119860

1 119860

2 119860

119899)

(1198861 119886

2 119886

119899) ⊊ (119861

1 119861

2 119861

119899)

(44)

If 119861 ⊊ 119860 then 119861119894⊊ 119860


least 119887119894isin 119861

119894such that 119887

119894notin 119860

119894 So 119887

119894 sube 119861

119894and 119887

119894 ⊊ 119860

119894 From

here we have (1198871 119887

2 119887

119899) sube (119861

1 119861

2 119861

119899) = 119861

and (1198871 119887

2 119887

119899) ⊊ (119860

1 119860

2 119860

119899) = 119860 If there


119894and 119861



119894 119886

119894notin 119861

119894and 119887

119894isin 119861

119894

119887119894notin 119860


119894 sube 119860

119894 119886

119894 ⊊

119861119894and 119887

119894 sube 119861

119894 119887

119894 ⊊ 119860


have (1198861 119886

2 119886

119899) sube 119860 (119886

1 119886

2 119886

119899) ⊊ 119861 and

(1198871 119887

2 119887

119899) sube 119861 (119887

1 119887

2 119887

119899) ⊊ 119860 So 119868R119899 is



(119868R119899

)119904

= 0

cup (1198831 119883

2 119883

119899) 119883

1 119883

2 119883

119899isin (119868R)

119904

(45)



119868R119899

(119868R119899

)119904 and (119868R119899


119903 minus dim (R119899

) = 119899

119904 minus dim (R119899

) = 0

119903 minus dim (119868R119899

) = 119899

119904 minus dim (119868R119899

) = 119899

119903 minus dim (119868R119899

)119904= 0

119904 minus dim (119868R119899

)119904= 119899


119903 minus dim (119868R119899

)119903= 119899

119904 minus dim (119868R119899

)119903= 0

(46)


)119904cup


and 1198862= (0 [2 4]) of 119860

119904 The set 119886



1and 120582

2satisfying the

inclusion (0 0) sube 12058211198861+ 120582





119868119904 = 119883 = (119883119899) isin 119868R

infin

(119883119899) = (119883

1 119883

2 119883

119899 )

isin 119868Rinfin

(47)


2

119883119899) isin 119868R119899 and 119884 = (119884

1 119884

2 119884



119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899 ) (48)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899 ) (49)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (50)


119889 (119883 119884) =

infin

sum

119894=1

1

2119894[ℎ (119883

119894 119884

119894)

1 + ℎ (119883119894 119884

119894)] (51)



119868119897infin= 119883 = (119883

119899) isin 119868R

infin

(119883119899) bounded (52)




infin (53)




sup 1003817100381710038171003817119883119894 + 1198841198941003817100381710038171003817119868R le sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R+10038171003817100381710038171198841198941003817100381710038171003817119868R

le sup 10038171003817100381710038171198831198941003817100381710038171003817119868R + sup 1003817100381710038171003817119884119894

1003817100381710038171003817119868R

le infin

sup 10038171003817100381710038171205821198831198941003817100381710038171003817119868R le sup 120582 1003817100381710038171003817119883119894

1003817100381710038171003817119868R le 120582 sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R

le infin

(54)

119868119897infin


119883 = sup 10038171003817100381710038171198831198991003817100381710038171003817119868R 119899 isin N (55)


120598and 119883

120598 le 120598 for every




119894+ 119883



119883119894le 119884


Denote

1198681198880= 119883 = (119883

119899) isin 119868R

infin

(119883119899) 997888rarr 0 (56)


119899)


lim119899rarrinfin

10038171003817100381710038171198831198991003817100381710038171003817119868R= 0 lArrrArr 119883 isin 119868119888

0 (57)



(50)The space 119868119888

0with the norm

119883 = sup119894isinN

10038171003817100381710038171198831198941003817100381710038171003817119868R (58)


defined by

1198681198972= 119883 = (119883

119899) isin 119868R

infin

infin

sum

119894=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868Rbounded (59)



119889 (119883 119884)

= inf 119903 ge 0 119883 sube 119884 + 119878119903(120579) 119884 sube 119883 + 119878

119903(120579)

(60)

where 119878119903(120579) = 119883 isin 119868119897

2 119883 le 119903


119883 = (

infin

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(61)


ear space with (61)






(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)




2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)


119889 (119883119899

119894 119883

119898

119894) le 120598 (64)


119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)


0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)



119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)


119894=1119883

1198942

le infin



119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897




2hArr sum

infin

119899=1119883

1198992


119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)


119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)


119883119894


119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)


since 1198681198972is a




0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904





([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)


0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)



dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119903 minus dim (119868R119899

)119903= 119899

119904 minus dim (119868R119899

)119903= 0

(46)


)119904cup


and 1198862= (0 [2 4]) of 119860

119904 The set 119886



1and 120582

2satisfying the

inclusion (0 0) sube 12058211198861+ 120582





119868119904 = 119883 = (119883119899) isin 119868R

infin

(119883119899) = (119883

1 119883

2 119883

119899 )

isin 119868Rinfin

(47)


2

119883119899) isin 119868R119899 and 119884 = (119884

1 119884

2 119884



119883 + 119884 = (1198831+ 119884

1 119883

2+ 119884

2 119883

119899+ 119884

119899 ) (48)


120572 sdot 119883 = (120572 sdot 1198831 120572 sdot 119883

2 120572 sdot 119883

119899 ) (49)


119883 le 119884 lArrrArr 119883119894le 119884

1198941 le 119894 le 119899 (50)


119889 (119883 119884) =

infin

sum

119894=1

1

2119894[ℎ (119883

119894 119884

119894)

1 + ℎ (119883119894 119884

119894)] (51)



119868119897infin= 119883 = (119883

119899) isin 119868R

infin

(119883119899) bounded (52)




infin (53)




sup 1003817100381710038171003817119883119894 + 1198841198941003817100381710038171003817119868R le sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R+10038171003817100381710038171198841198941003817100381710038171003817119868R

le sup 10038171003817100381710038171198831198941003817100381710038171003817119868R + sup 1003817100381710038171003817119884119894

1003817100381710038171003817119868R

le infin

sup 10038171003817100381710038171205821198831198941003817100381710038171003817119868R le sup 120582 1003817100381710038171003817119883119894

1003817100381710038171003817119868R le 120582 sup 1003817100381710038171003817119883119894

1003817100381710038171003817119868R

le infin

(54)

119868119897infin


119883 = sup 10038171003817100381710038171198831198991003817100381710038171003817119868R 119899 isin N (55)


120598and 119883

120598 le 120598 for every




119894+ 119883



119883119894le 119884


Denote

1198681198880= 119883 = (119883

119899) isin 119868R

infin

(119883119899) 997888rarr 0 (56)


119899)


lim119899rarrinfin

10038171003817100381710038171198831198991003817100381710038171003817119868R= 0 lArrrArr 119883 isin 119868119888

0 (57)



(50)The space 119868119888

0with the norm

119883 = sup119894isinN

10038171003817100381710038171198831198941003817100381710038171003817119868R (58)


defined by

1198681198972= 119883 = (119883

119899) isin 119868R

infin

infin

sum

119894=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868Rbounded (59)



119889 (119883 119884)

= inf 119903 ge 0 119883 sube 119884 + 119878119903(120579) 119884 sube 119883 + 119878

119903(120579)

(60)

where 119878119903(120579) = 119883 isin 119868119897

2 119883 le 119903


119883 = (

infin

sum

119894=1

10038171003817100381710038171198831198941003817100381710038171003817

2

119868R)

12

(61)


ear space with (61)






(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)




2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)


119889 (119883119899

119894 119883

119898

119894) le 120598 (64)


119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)


0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)



119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)


119894=1119883

1198942

le infin



119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897




2hArr sum

infin

119899=1119883

1198992


119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)


119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)


119883119894


119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)


since 1198681198972is a




0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904





([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)


0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)



dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(

infin

sum

119899=1

1003817100381710038171003817119883119899 + 1198841198991003817100381710038171003817

2

119868R)

12

le (

infin

sum

119899=1

(10038171003817100381710038171198831198991003817100381710038171003817119868R+10038171003817100381710038171198841198991003817100381710038171003817119868R)2

)

12

le (

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

+ (

infin

sum

119899=1

10038171003817100381710038171198841198991003817100381710038171003817

2

119868R)

12

lt infin

120582119883 = (

infin

sum

119899=1

10038171003817100381710038171205821198831198991003817100381710038171003817

2

119868R)

12

= 120582(

infin

sum

119899=1

10038171003817100381710038171198831198991003817100381710038171003817

2

119868R)

12

lt infin

(62)




2 If

119883119899

= (119883119899

1 119883

119899

2 119883

119899

3 ) (63)


119889 (119883119899

119894 119883

119898

119894) le 120598 (64)


119883119899

119894le 119883

119898

119894+ 119860

1120598

119894

119883119898

119894le 119883

119899

119894+ 119860

2120598

119894

(65)


0such that

10038171003817100381710038171003817119860119895120598

119894

10038171003817100381710038171003817119868Rle 120598 (66)



119883119899

119894997888rarr 119883

119894119894 = 1 2

119883 = (1198831 119883

2 )

(67)



119883119899

119894le 119883

119894+ 119861

1120598

119894

119883119894le 119883

119899

119894+ 119861

2120598

119894

10038171003817100381710038171003817119861119895120598

119894

10038171003817100381710038171003817119868Rle 120598

120484

(68)


119883119899

le 119883 + 1198611120598

119883 le 119883119899

+ 1198612120598

(69)

but1003817100381710038171003817100381711986111989512059810038171003817100381710038171003817≰ 120598 (70)


119894=1119883

1198942

le infin



119883119899

997888rarr 119883 997904rArr 119883119899

119894997888rarr 119883

119894(71)

but

119883119899

119894997888rarr 119883

119894999424999426999456 119883

119899

997888rarr 119883 (72)

for every119883119899 = (1198831198991 119883

119899

2 )

Example 47 1198681198972 119868119888

0 and 119868119897




2hArr sum

infin

119899=1119883

1198992


119865119883= 119884 isin (119868119897

2)119903 119884 le 119883 (73)


119894isin R and 119883

119894isin 119868R

such that

119884119894le 119883

119894(74)


119883119894


119883119894= sup119865

119883119894= 119884

119894isin R 119884

119894le 119883

119894 (75)


since 1198681198972is a




0 for 119861 = ([minus1 1] 0 0 ) isin (1198681198880)119904





([1 2] 0 0 ) (0 [1 2] 0 0 )

(0 0 [1 2] 0 )

(76)


0 = (0 0 )

sube 1205821([1 2] 0 0 ) + 120582

2(0 [1 2] 0 0 ) + sdot sdot sdot

+ 120582119896(0 0 [1 2] 0 )

= (1205821[1 2] 120582

2[1 2] 120582

119896[1 2] 0 0 )

lArrrArr 1205821= 120582


119896= 0

(77)



dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











dim119883 = infin


119883

= (119868119897infin)119904

cup (0 0 0 119896 0 0 0 119897 0 0 ) 119896 119897 isin R

(78)

Theorem 52 Let 1198641= (1 0 0 0 ) 119864

2= (0 1


0

and 119868119897119901for (119901 le 1)


0 the


119896)infin

119896=1isin 119868119888

0and 119885 =

(1198831 119883

2 119883

119899 ) isin (119868119888

0)119903for every 119885 isin 119865

119884 We choose

1205721= 119883

1 120572

2= 119883

2 120572

119899= 119883


1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

119883119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

=1003817100381710038171003817(1198831 1198832 119883119899 )

minus (1198831 119883

2 119883

119899 0 0 )

1003817100381710038171003817

=10038171003817100381710038170 0 119883119899+1 119883119899+2

1003817100381710038171003817

= sup119896ge119899+1

10038161003816100381610038161198831198961003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(79)

Since

ℎ(119885

119899

sum

119896=1

120572119896119864119896) le

1003817100381710038171003817100381710038171003817100381710038171003817

119885 minus

119899

sum

119896=1

120572119896119864119896

10038171003817100381710038171003817100381710038171003817100381710038171198681198880

(80)



119885 =

infin

sum

119896=1

120572119896119864119896

119885 =

infin

sum

119896=1

120573119896119864119896

(81)


0)119903 we get

0 = 119885 minus 119885 =

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=1

120572119896119864119896minus

infin

sum

119896=1

120573119896119864119896

1003817100381710038171003817100381710038171003817100381710038171003817


1003817100381710038171003817


1003816100381610038161003816120572119896 minus 1205731198961003816100381610038161003816

(82)

Hence we find 120572119896= 120573


theorem



Competing Interests


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







