Topologies Associated with the Compact Open Topology on (U)

Topologies Associated with the Compact Open Topology on 퓗(U)Author(s): J. M. Ansemil and S. PonteSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 82A, No. 1 (1982), pp. 121-128Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20489145 .

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TOPOLOGIES ASSOCIATED WITH THE COMPACT OPEN TOPOLOGY ON JfiU)

By

J. M. Ansemil and S. Ponte

Facultad de Matem?ticas, Universidad de Santiago, Spain

(Communicated by D. J. Judge, m.rj.a.)

[Received 22 May 1981. Read 14 December 1981 Published 23 July 1982.1

ABSTRACT

Let U be a balanced open subset of a complex locally convex space E and let JfiU) be the space of holomorphic functions on U. Noverraz in Proc. R. Ir. Acad. 11A has considered various topologies associated with rw on JfiU). In this paper we consider a similar situation starting from r0 instead of rw; we give general examples (and specific counterexamples) of complex locally convex spaces for which rs is (and is not) the barrelled or bornological topology associated with r0 on JfiU).

Introduction

Let U be an open subset of a complex locally convex space E and let us consider on the

space JîU) of holomorphic functions on U the following topologies:

the compact open topology r0;

the rw topology defined by the seminorms on Jf iU) ported by the compact subsets of U; a seminorm p on Jf ((/) is ported by the compact subset K of U if for each open subset V

of U containing K there exists C(F) > 0 such that

/K/KC(K> sup I/Ml xeV

for all / in JfiU); the r5 topology defined by the seminorms p on Jf iU) such that for each countable increasing open cover (?/?) of ?/, there exist C > 0 and a positive integer n0 such that

X/K sup l/MI xeUm

for all/in Jf(?)

r0 < x < rs and rs is ultrabornological.

If we consider the spaces ?^(rt?), n G N, of ?-homogeneous continuous polynomials, we

can define in a similar way the topologies r0, rw and rs ona?inE). If we denote these topologies

Proc. R. Ir. Acad. Vol. 82a, No. 1, 121-128 (1982)



122 Proceedings of the Royal Irish Academy

by TQ(d?(nE)), im(^(nE)) and x3(t?(nE)) respectively and if U is balanced we have

= rJ^(nE))

= r^(rtE))^T?

= T0m?E)),T(ti

^(nE) &(?E) ^(nE)

and xJ(^(nE)) is ultrabornological [3], [7]. Let G be a locally convex HausdorfT space with a topology x. It is known [18] that it is

possible to associate with r a bornological (respectively, barrelled, infrabarrelled and

ultrabornological) topology, which is the locally convex inductive limit of all bornological

(resp. barrelled, infrabarrelled and ultrabornological) topologies stronger than r. We will call rb

(resp. tr Tj and r^) the bornological (resp. barrelled, infrabarrelled and ultrabornological)

topology associated with r on G. It is clear that z? < xb < x?? and rt < xt < x^ In this paper we show that, if E is a fc-space, then x? is the bornological topology

associated with the compact open topology on Jf(U) and that if U is a balanced open subset of a metrizable or DF space, then x6 is also the barrelled topology associated witr

r0 on Jf(U). Moreover, we give examples where x6 is not the bornological, barrelled or

ultrabornological topology associated with r0 on Jf(lf).

1. The bornological topology associated with r0 on Jf(U)

In this section E stands for a complex locally convex HausdorfT space and U a non-void

open subset of E.

Proposition 1. If U is a k-space, then all null sequences in (Jf(U), r0) are locally bounded.

Proof. Let (fn) c Jt(U) be a sequence such that/? - 0 in (Jf(U), to) and let z0 e U. As

lim f?(z0) = 0

? -+00

there exists C > 0 such that

|/w(z0)|<C

for all n. Let us write

Q = {z G U : |/?(z) | < C + 1 for all n).

We will show that Q is an open subset of U. U is a espace, hence ?2 is open if and only if for

each compact subset K of U, K n ?2 is an open subset of A". Let K c ?/ be compact and z e

tf n ?2. As/, - 0 in (Jf(L0, r0) to^ is an no e N such *at

sup|/,(x)|<C+l xeK

for all n > ?0. On the other hand, there exists a neighbourhood *V2 of z in U such that

sup|/?(x)|<C+l

for ? ? 1, 2, . . ., ?0. Hence



Ansemil and Ponte?Topologies associated with r0 on Jf\U) 123

sup |/B(x)|<C+ 1 xevznK

for every n G N, and

vz n K <z n n K.

Remark 1. If ? is a ?-space, then every open subset of E is a /espace [8, Prop. 8.4].

Theorem 1. If U is a k-space, then z? is the bornological topology associated with r0 on

JfiU).

Proof. Proposition 1 shows that each r0?-null sequence of JfiU) is locally bounded, hence

r^-bounded [3]. Therefore, the identity mapping

(Jf(U), r0)b) - iJfiVl r?)

is continuous and, as consequence, z0jb > z?\ then zs =

z0?.

Corollary 1. If U is a hemicompact k-space, then z? =

zQ on JfiU).

Proof. With these conditions iJtiU), r0) is a Fr?chet space, hence bornological. Then

\b =

ro anc*, by Theorem 1, z? =

r0?; therefore rs =

z0 on JîU).

Remark 2. ia) Let E be an infinite dimensional Banach space; then E is a ?-space which is

not hemicompact (Baire's theorem) and zs =? r0 on Jf iE) [1]. Later on we will see an example of a hemicompact non ?-space such that z? ̂ r0 on JfiU).

ib) If F is a Fr?chet space, E ?

(F\ r0) is called a DFC space (see f 10], [111 and [151). Then E is a /espace (Banach-Dieudonn? Theorem) and each pseudoconvex open subset of E

is hemicompact [ 19]. If F is a separable Fr?chet space, every open subset of E = (F, Tq) is

hemicompact ?15]. If F is a Fr?chet space, E = (?^(WF), Tq) is a hemicompact ?space (the n

homogeneous polynomials (see [14]) of a fundamental sequence of 0-neighbourhoods in F

are a fundamental sequence of compact subsets of E and the Banach-Dieudonn? Theorem for

homogeneous polynomials can be applied).

Corollary 2. If U is a k-space, then iJfiU), zj is bornological if and only if it is barrelled.

Proof. If (Jf(t/), zj is barrelled we have zw =

zs [ 17], hence (JfiU), r?) is bornological.

Conversely, if (Jf (?/), z ) is bornological we have zm = z J,. By Theorem 1 z J)

= z?; therefore

tw =

z? and iJtiU), rj is barrelled.

Remark 3. If E is metrizable (hence ?-space) and there does not exist a continuous norm on

F, then (Jf(?), rw) is not bornological [3], hence not barrelled.

2. The barrelled topology associated with r0 on Jf(U)

In this section U stands for a balanced open subset of a locally convex space E.

We recall the following definition: a sequence (<7n) of subspaces of a locally convex space




G is a Schauder decomposition of G if any x in G can be written in a unique way as x = L xn with xn in Gn for each n G N, and the mappings

un : x =-- ? xn ->

xn, x e G

are continuous.

It is well known [3] that (^("E)f x)) is a Schauder decomposition for (Jt(U), r), where x =

r0, x or xs. To show that the sequence (iJ/(nE)) with the corresponding induced topologies is

also a Schauder decomposition for the barrelled topology associated with r0 on Jf(U), we

prove the following.

Lemma 1. Let G be a locally convex Hausdorff space with a topology x and F a

complemented subspace of G. Then r,?F =

(r[F),.

Proof. Let u : (G, x) -> (F, T[f) be a continuous projection. As xt > r,

u : (G, r,) - (F, xlF)

is continuous, and by Theorem 1.4.1 of [18]

u : (G, r,) -

(F, (r|F),)

is also continuous. Hence, as u is a projection

% > (Vf On the other hand, the inclusion

/ : (F, r,p -

(G, r)

is continuous; therefore so is

/ : (F, (x]F)t) - (G, r).

By Theorem IA Lof [18]

i : (F, (r,pf) - (G, r,)

is continuous. Hence

( Vi > V Then r^

- (r^),.

Remark 4. A similar proof shows that for any locally convex Hausdorff spaces G and H

(G + //), =

G, + //,.

Proposition 2. ((^("F), r0,)) /s a Schauder decomposition for (Jf(U), x0).

Proof. As x0t < x? we have

/= I ? */(0)in(^(W,rftl).

Since (^(nE), x0) is a complemented subspace of (Jf(i$, r^ Lemma 1 implies that (?P(nE),xQj (Jf(U)))

= (^(nE), r0 ,). Also by Lemma 1 the projections on ?j*("?) are continuous for all /?

and this completes the proof.



Ansemil and Ponte?Topologies associated with r0 on Jr"iU) 125

Theorem 2. zs is the barrelled topology associated with r0 on Jf iU) if and only if z is

the barrelled topology associated with r0 on ^{nE) for each n G X.

Proof. Suppose z6 =

z04 on Jf\U). Since zb ?

rw on ̂ inE) and since we have seen in the

proof of Proposition 2 that (J*("E), z0t (t?("E)) = (^(*F), r0/ (.#(?7))), this implies that zw =

r0, on ,

A"?).

Conversely, if rw =

r0/ on ^("F), then (^(nF), zj) is a Schauder decomposition for two

barrelled topologies on Jf(U), z04 and r?, and by a result of Noverraz [ 171 we have z? =

r0 , on

X(l/).

Remark 5. (a) If ra, =

r? on Jf(?), then r^ -

r? on Jt(U).

ib) The above Theorem shows that for balanced open subsets the questions (/>) and (c) of

[4] are equivalent.

Corollary 3. If E is metrizable, then r0i/ ?

z? on M\U).

Proof. As E is metrizable rw =

z0t on^("F) for each n e S [5].Hence z0i =

z? on Jf\U).

Coroflary 4. If E is metrizable, then iJf(U), z^ and iJfiU), z3) are complete (see also 13]

and [141).

Proof. As E is metrizable it is a ?-space, hence (JfiU), Zq) is complete. Then, as r? =

z0j on

Jf(LO, iJfiJJ), z?) is complete [18, Corollaire 1.6.4]. On the other hand, it is known 131 that for

F metrizable (JT(?/), zw) is complete if and only if iîU), r?) is complete.

Proposition 3. If E is a DF space, then the strong topology ? on ,J/inE) (see [71) coincides

with zw on înE)for each n e S.

Proof. We have ? < rw (17]). Since F is a DF space, iînE), ?) is bornological, hence the

identity map iî?E), ?)

- iî?E), zj

is continuous if every null sequence on 0^("F), ?) is equicontinuous (equicontinuous subsets

of ̂ inE) are locally bounded, hence rebounded [71). If/7=1 this is true by the definition of a

DF space. We consider now the case n ? 2; the general case is similar. Let iP) c ?^(2F) be a

sequence such that Pj

- 0 in (^(2F), ?). For each y G N let A} G i^(2F) such that

Pjix) =

Ajix,x) for x G F. 04,) is a bounded subset of i,J/ i2E\?) and it is the union

of a sequence of equicontinuous sets, hence iA? is equicontinuous [9, Prop. 4.3.3],

Hence (Fy) is an equicontinuous subset of f^(2F).

Corollary 5. If E is a DF space, then z0t =

zs on Jf\U).

Proof. Since the /^-neighbourhoods of 0 in & inE) are r0-barrels, we have ? < z0t < zw on

înE)\ as ? =

zw ontyinE) we have r^ôn ^(*F). Thus, t0, =

r? on jf(U) by Theorem 2.

Remark 6. If F is a fully nuclear space with a basis, then r? =

zQ4 on JfiU) for every

polydisc U of F [41. If F is a strict inductive limit of Fr?chet nuclear spaces and F admits a




continuous norm, then xs =

r^ on Jt(U) [7] (if E = EEn where each En is a Fr?chet nuclear

space with a continuous norm, then E satisfies the above conditions).

Proposition 4. If F is a non-complete Montel space [13], then x0t =? x on F'; hence

x0tt * xson Jf (If).

Proof. (F\ /?) = (F\ x0) is barrelled, therefore x0j

= r0 on F'. Assume that we had r0f

= rw

on F'; since (F', # is bornological, F - ((F', #\ j?

= ((?', rj\ $ would be complete.

3. Other topologies associated with r0 on Jf(?/)

In this section we study the ultrabornological and infrabarrelled topologies associated with

t0 on r#XU), where U stands for a balanced open subset of E.

Theorem 3. xh is the ultrabornological topology associated with x0 on Jf(U) if and only if xw is the ultrabornological topology associated with x0 on ^(nE)for each n e IN.

Proof. Analogous to Theorem 2.

Remark 7. If xQ4 ?

xs on JfifJ) (for example for metrizable or DF spaces), then xM =

x? on

,W\U) because x0tl < xm < xs.

Proposition 5. If F is an infinite dimensional reflexive Banach space and E = (F, o) (o denotes the weak topology with respect to the duality (F, F'?, then '(?$(nE), x<) is

ultrabornological and r0 =? rw on J/(nE).

Proof. ^(jF) = F and (^('F), t0) = (F ', || ||') (Alaoglu-Bourbaki Theorem). Then

(?^(]F), r0) is ultrabornological. On the other hand we have

&m,rJ = \imWEM \\v) ?*

v

where V ranges over all a-neighbourhoods of zero in F [7]. For each a-neighbourhood V of zero in F, ??(lEv) is a finite dimensional space. Thus (&(lE\ xj

= (F)*. As F is

reflexive and (&?E), r0) = (F', ||' ||') we have (??('?), r0)'

- F. Since (F')* * F, we have

shown that r0 ̂ rw on ^(!F).

Corollary 6. If F is an infinite dimensional reflexive Banach space and E = (F, ex),

then x0tUb * r? (/lence r0i/ *= r?) ow Jf(i7).

Remark 7. Corollary 6 gives an example of a hemicompact space which is not k

space.

Proposition 6. r0? =

x0J on Jt(U) if and only if ' r0?

= r0/ on i^(nE)for each n g N.

Proof. A proof similar to Lemma 1 shows that (&(nE), x0tb) = (?*(*F), r06(Jf (t/)))

and ( &(nE), x0J) = (^(nF), r0i/ (Jf(l/))). Hence, if tw

= x0J on>(?/), we have ra6

= rw

on &(nE) for every w G N.



Ansemil and Ponte?Topologies associated with z0 on Jf iU) 127

Conversely, let p be a z0Jb-continuous seminorm on JfiU). We may suppose that

pV)=1p (-?-*/(0)) n = Q \ ft. /

for each/in .W\U) 13]. Let us write

Pk(f) = I P (~?7 */?? ) >/? *"?/> n=0

for k G N. Since

{/GJf(W:p(/)<l}= fi {/eJf(t/):^/)<l}

is a r0>i-closed balanced subset of JfiU) which absorbs the rebounded sets, p is

r0?-continuous.

Corollary 7. If iJ\U), z0) is complete, then z0? =

r? whenever z0t =

z? on ,J^(U).

Proof. If (Jf(LO, r0) is complete, (Jf(10, r0J)is complete 118, Corollaire 1.6.4). Hence zQJ =

r0j/ on Jf(?/). Then, if z0j =

r? on Jf (?/), we have r0j/ =

zs on Jf(?/). But r0i/ < r0J} < r^ Then

zQJb =

r, on Jf(L/).

Remark 8. If F is a DF (or a fully nuclear space with basis) then r0? =

r? on Jf\U) for every balanced (resp. polydisc) open subset U of F when iJfiU)-, z?) is complete (see |6|).

ACKNOWLEDGEMENTS

We thank Professors S. Dineen and J. Mujica for suggestions given during the

preparation of this paper.

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Topologies Associated with the Compact Open Topology on (U)