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JOURNAL OF c© 2006, Scientific Horizon

FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net

Volume 4, Number 1 (2006), 1-6

Characterizations of inner product spaces by

orthogonal vectors

Marco Baronti and Emanuele Casini

(Communicated by Jurgen Appell)

2000 Mathematics Subject Classification. 46C15, 46B99, 41A65.

Keywords and phrases. Orthogonality, Characterizations of inner product

spaces.

Abstract. Let X be a real normed space with unit closed ball B . We prove

that X is an inner product space if and only if it is true that whenever x, y are

points in ∂B such that the line through x and y supports√

22

B then x ⊥ y in

the sense of Birkhoff.

1. Introduction

Several known characterizations of inner product spaces are available inthe literature [1]. This paper concerns a new characterization by means oforthogonal vectors. Let X be a real normed space (of dimension ≥ 2) andlet SX denote its unit sphere. If x, y ∈ SX , we set:

k(x, y) = inf{‖tx + (1 − t)y‖ : t ∈ [0, 1]}

moreover if x, y ∈ SX then x ⊥ y denotes the orthogonality in the senseof Birkhoff [1] i.e. ‖x‖ ≤ ‖x + λy‖ ∀λ ∈ R . In this paper we consider

2 Characterizations of inner product spaces

the following properties

(P1) : x, y ∈ SX x ⊥ y ⇒ k(x, y) =‖x + y‖

2

(P2) : x, y ∈ SX k(x, y) =√

22

⇒ x ⊥ y

We prove that these properties characterize inner product spaces. Thedefinition of the constant k(x, y) is suggested by the following result byBenitez and Yanez [4]:{

x, y ∈ SX k(x, y) =12⇒ x + y ∈ SX

}⇔ X is an inner product space.

It is easy to prove that every inner product spaces X satisfies the properties(P1) and (P2). Moreover from the particular structure of (P1) and (P2) wecan suppose that X is to be a 2-dimensional real normed space.

2. Some preliminary results

We start with some preliminary observations on the constant k(x, y).If x, y ∈ SX and x ⊥ y we set:

μ(x, y) = supλ≥0

1 + λ

‖x + λy‖

andμ(X) = sup{μ(x, y); x, y ∈ SX x ⊥ y }.

These constants were introduced in [6] and in [6] and [3] the followingresults were proved:

μ(X) ≥√

2 for any space X

andμ(X) =

√2 ⇔ X is an inner product space

Let x, y ∈ SX , x ⊥ y, we have

μ(x, y) = supλ≥0

1 + λ

‖x + λy‖ = sup0≤t<1

1 + t1−t

‖x + t1−ty‖

= sup0≤t≤1

1‖tx + (1 − t)y‖

=1

k(x, y)

M. Baronti and E. Casini 3

and so

(1) μ(X) =1

inf {k(x, y) : x, y ∈ SX x ⊥ y}From known results on the constant μ [3], [4] we obtain the following results:

inf {k(x, y) :x, y∈SX x ⊥ y}≤√

22

(2)

inf {k(x, y) :x, y∈SX , x ⊥ y}=√

22

⇔ X is inner product space.(3)

Theorem 1. If X fulfills (P1) then X is an inner product space.

Proof. Let x, y ∈ SX with x ⊥ y. From (P1) we have k(x, y) = ‖x+y‖2 .

We define g : R → R by g(t) = ‖tx + (1 − t)y‖. g is a convex function andg(1

2 ) ≤ g(t) ∀ t ∈ [0, 1] . So we have g(12 ) ≤ g(t) ∀ t ∈ R. Hence

‖x + y‖2

≤∥∥∥∥y + t(x − y) +

x + y

2− x + y

2

∥∥∥∥ =∥∥∥∥x + y

2+

2t − 12

(x − y)∥∥∥∥ .

So||x + y||

2≤

∥∥∥∥x + y

2+ λ(x − y)

∥∥∥∥ ∀λ ∈ R

that is equivalent tox + y ⊥ x − y.

Therefore ∀ x, y ∈ SX x ⊥ y ⇒ x + y ⊥ x − y that is a characteristicproperty of inner product spaces [2]. �

3. Main result

To prove our main result we start with the following Lemma

Lemma 2. Let X be a 2-dimensional real normed space and let x be inSX . Let γ be one of the two oriented arcs of SX joining x to −x. Thenthe function y ∈ γ �→ k(x, y) is nonincreasing.

Proof. Let y and y ∈ γ and we suppose that y belongs to the part ofthe arc γ joining −x to y. We will prove that

k(x, y) ≤ k(x, y).


We can suppose y = ±x so there exist a, b ∈ R such that y = ax + by.

From the assumption about y we have a ≥ 0 and b ≥ 0 and so

‖y‖ = 1 = ‖ax + by‖ ≤ a + b .

If a + b = 1, then y = ax + (1 − a)y hence x, y and y are collinear andthis implies:

1 = k(x, y) = k(x, y) .

Now we suppose a + b > 1 and k(x, y) > k(x, y). Then there existst1 ∈ (0, 1) such that ||t1x + (1 − t1)y|| < k(x, y) ≤ 1. Let

λ =1

t1(1 − a − b) + a + b.

It is easy to verify that λ ∈ (0, 1). Let

qλ = λ(t1x + (1 − t1)y)

=t1

t1(1 − a − b) + a + bx +

(1 − t1)(ax + by)t1(1 − a − b) + a + b

=(t1 + a − at1)

t1(1 − a − b) + a + bx +

(1 − t1)bt1(1 − a − b) + a + b

y .

So qλ belongs to the segment joining x to y and

‖qλ‖ = λ‖t1x + (1 − t1)y‖ ≥ k(x, y) > ‖t1x + (1 − t1)y‖ .

So λ > 1. This contradiction implies the thesis. �

Theorem 3. If X fulfills (P2) then X is an inner product space.

Proof. We suppose that X is not an inner product space. So by usingresults (2) and (3) there exist x, y ∈ SX , x ⊥ y such that k(x, y) <

√2

2 .Let γ be one of the two oriented arcs of SX joining x to −x. Let z1 andz2 be in γ such that x ⊥ z1 , x ⊥ z2 and the part of γ joining z1 to z2 isthe arc containing all points z such that x ⊥ z and moreover we supposethat z1 belongs to the part of γ joining x to z2 ; we will write z1 ∈ xz2.Note that y ∈ z1z2. Then using a result by Precupanu [7] we can supposethat there exists a sequence (xn) such that:

1. xn ∈ xz1

2. xn = x3. xn → x

4. xn are smooth.

M. Baronti and E. Casini 5

Let yn be in γ such that xn ⊥ yn. From the monotony of the Birkhofforthogonality it follows that yn ∈ −xz2 Let ε be a positive number suchthat

k(x, y) + ε <

√2

2.

From Lemma 2 and the continuity of the function k(· , y) there exists n ∈ N

such that for n > n

k(xn, yn) ≤ k(xn, y) ≤ k(x, y) + ε <

√2

2.

Now if we fix n0 > n, we have

k(xn0 , yn0) <

√2

2; xn0 ⊥ yn0 ; xn0 is smooth .

The continuous function k(xn0 , ·) assumes values between k(xn0 , yn0) <√

22

and k(xn0 , xn0) = 1. So there exists y∗ = yn0 , y∗ ∈ xn0yn0 , k(xn0 , y∗) =√

22 . From (P2) we have xn0 ⊥ y∗ . But xn0 is a smooth point and so

y∗ = yn0 . This contradiction completes the proof. �

References

[1] D. Amir, Characterizations of Inner Product Spaces, Birkhauser Verlag,Basel, 1986.

[2] M. Baronti, Su alcuni parametri degli spazi normati, B.U.M.I.(5), 18-B(1981), 1065–1085.

[3] C. Benitez and M. Del Rio, The rectangular constant for two-dimensional spaces, J. Approx. Theory, 19 (1977), 15–21.

[4] C. Benitez and D. Yanez, Two characterizations of inner productspaces, to appear.

[5] R. C. James, Orthogonality in normed linear spaces, Duke Math. J.,12 (1945), 291–302.

[6] J. L. Joly, Caracterisations d’espaces hilbertiens au moyen de laconstante rectangle, J. Approx. Theory, 2 (1969), 301–311.

[7] T. Precupanu, Characterizations of hilbertian norms, B.U.M.I.(5), 15-B (1978), 161–169.


Dipartimento di Ingegneria della Produzione e Metodi e Modelli MatematiciUniversita degli studi di GenovaPiazzale Kennedy, Pad. D16129 GenovaItaly(E-mail : [email protected])

Dipartimento di Fisica e MatematicaUniversita dell’InsubriaVia Valleggio 1122100 ComoItaly(E-mail : [email protected])

(Received : July 2004 )

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