METRIZABILITY OF SPACES OF LIPSCHITZ FUNCTIONS (Researches on isometries from various viewpoints)

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METRIZABILITY OF SPACES OF LIPSCHITZ FUNCTIONS

A.JIMÉNEZ‐VARGAS

1. INTRODUCTION

Let

_{\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)}

bethe linearspaceof all scalar‐valued

Lipschitz

functions

vanishing

at0on anormedspace

E. Let $\tau$ bea

locally

convex

topology

on

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

such that $\tau$_{0} \leq $\tau$ \leq $\tau \delta$_, where $\tau$_{0} and $\tau$_{ $\delta$} denote the

compact‐open

topology

and the Nachbm‐C‐Couré

topology

on

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

.

We prove in this note that

_{(\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E), $\tau$_{0})}

is ametrizable space ifand

only

ifE has finite dimension.

Motivated

by

a

positive

answerinthe

setting

of

holomorphic

mappings,

the

following

question

israised: Is

ittruethat

_{(\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E), $\tau$)}

is metrizable

_only

if E is finite‐dimensional? 2. PRELIMINARIES

Let E beanormedspaceand let

_{\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)}

denote the linearspaceof all

Lipschitz

mappings

f

from E into \mathrm{K} for which

_f(0)=0

. We refer the readertoWeaversbook

[6]

forthe basic

theory

of

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

.

Let X be a

_topological

spaceand let

C(X)

be the linearspaceof all continuous

mappings

from X into K. Werecall the

_following

_topologies

on

C(X)

.

The

_{compact‐open}

_topology

on

C(X)

isthe

locally

convex

topology generated by

the seminorms

|f|_{K}=\displaystyle \sup_{x\in K}|f(x)|, f\in C(X)

,

where K variesoverthe

_family

of all_{compact subsets}of X.

Aseminormpon

C(X)

is

_porte.

\mathrm{d}

by

the_{compact subset}K of X iffor_everyopen

neighborhood

V of K in

X,there isaconstant

c_{V}>0

such that

p(f)\displaystyle \leq c_{V}\sup_{x\in V}|f(x)|

for all

f\in C(X)

. The Nachbin

topology

on

C(X)

isthe

locally

convex

topology generated by

theseminormson

C(X)

whichare

ported by

the

_compact

subsetsofX.

TheNachbin‐Couré

topology

on

C(X)

is the

locally

convex

topology generated by

the seminormsp on

C(X)

which

satisfy

the

following

property:

for each

_increasing

countableopencover

\{V_{n}\}_{n\in \mathrm{N}}

ofX,there

arem\in \mathrm{N}and

c_{m}>0

suchthat

_{p(f)\displaystyle \leq c_{m}\sup_{x\in V_{m}}|f(x)|}

for all

_{f\in C(X)}

.

We will denote

by

$\tau$_{0},$\tau$_{ $\gamma$}and T $\delta$thecompact‐open

topology,

the

_{Nachbin‐ported topology}

and the Nachbin‐ Couré

_topology

on

C(X)

,or onany linear

subspace

of

C(X)

.

Nowweprovethe

following

result.

Theorem 2.1.

_If

E isaBanachspace,then

(\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E), $\tau$_{0})

ismetrizable

if

and

only if

E has

_finite

dimension.

Proof.

Suppose

that

_{(\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E), $\tau$_{0})}

is metrizable. Then there existsa_sequence

\{K_{n}\}_{n\in \mathrm{N}}

of

_compact

subsets

ofE,

containing

the

origin,

suchthat thesequenceofseminorms

|\cdot|_{K_{n}}

definesthe

topology

$\tau$_{0} on

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

.

We claim that there existsaconstantc>0 such that E is included in

_{\displaystyle \bigcup_{n\in \mathrm{N}}c\overline{ $\Gamma$}(K_{n})}

,where

\overline{ $\Gamma$}(K_{n})

denotes

the

_closed,

convex,balancedhullof

K_{n}

in E.

Indeed, given

x\in E_,it isclear that

I

definedon

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

isacontinuous seminormon

(\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E), $\tau$_{0})

, sotherearem\in \mathbb{N}and c>0suchthat

|f|_{\{x\}}

\leq c|f|_{K_{m}}

for all

f\in

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

. It follows that

|f(x)|

\leq

c|f|_{\overline{ $\Gamma$}(K_{7 $\gamma$})}

for all

f

\in

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

. Noticethat each

\overline{ $\Gamma$}(K_{n})

is

compact

by

theMazurtheorem. Since the dualspaceE'isasubset of

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

,wehave

|f(x)|

\leq c|f|_{\overline{ $\Gamma$}(K_{m})}

forall

f\in E'

.

By

the Hahn‐Uanach

separation

theorem,

weinfer thatxis in

c\overline{ $\Gamma$}(K_{m})

as wewanted. Since E isa

Bairespace,ourclaim

implies

that thereexists

p\in \mathbb{N}

such that

\overline{ $\Gamma$}(K_{p})

hasno

_empty

interior in E. Hence

thereisa

_compact

neighborhood

of0 in E and therefore E hasfinitedimension

_by

theRiesztheorem.

Conversely,

if E isfinite

_dimensional,

then

_{(C(E), $\tau$_{0})}

ismetrizable

_(see

the

_proof

of

_[3,

Theorem

_16.9])

and thereforesois

_{(\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E), $\tau$_{0})}

. \square

Theresultsonthe

_{metrizability}

ofspacesof

holomorphic

functions havean

interesting

history.

In

_1968,

Alexander

_[\mathrm{i}]

proved

the

_following

theorem for Banach spaceswith Schauder

basis,

whichwas

generalized

by

Chae

_(see

_[3,

Theorem

_16.10]):

If U isan opensubset ofan infinite dimensional BanachspaceE and

数理解析研究所講究録

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A.JIMÉNEZ‐VARGAS

$\tau$isa

_topology

onthespace

H(U)

of all

holomorphic

functionson U finer than the

topology

of

_pointwise

convergence, then

(H(U), $\tau$)

isnotmetrizable.

In

2007,

this theorem

_probably

motivatedAnsemil and

Ponte,

whosepaper

[2]

contains that if U isan open subsetofaninfinite‐dimensional

complex

metrizable

locally

convexspaceE, then

(H(U), $\tau$_{ $\gamma$})

isnot metrizable. This answereda

_question

stated

_by

_Mujica

in

_[5,

Problem

_11.9]

_thirty

years ago. It is known

that

_{$\tau$_{0}\leq$\tau$_{ $\gamma$}\leq$\tau$_{ $\delta$}}

on

H(U)

.

In

2009,

López‐Salazar

[4]

improved

this result

_showing

that if U isanopensubset ofa

complex

metrizable

locally

convexspaceEand $\tau$isa

locally

convex

_topology

on

H(U)

such that $\tau$_{0}\leq $\tau$\leq T $\delta$,then

(H(U), $\tau$)

is

ametrizablespaceif and

only

if Ehas finite dimension.

Theorem 2.1

suggests

totacklethe

_problem

onthe

_{metrizability}

of

\mathrm{L}\mathrm{i}\mathrm{p}_{0}(E)

_equipped

with other

_topologies,

withan

_approach

similartothat describedaboveforspacesof

holomorphic

functions.

REFERENCES

[1]

H._{Alexander, Analytic}functionsonBanachspaces,Thesis,UniversityofCalifornia,Berkeley,1968.

[2]

JoséM.Ansemil and SocorroPonte, Metrizabilityof spaces ofholomorphicfunctions with the Nachbintopology,J.Math.

Anal.Appl.334_(2007),no.2,1146−1151.

[3]

S. B.Chae,Holomorphyand CalculusmNormedSpaces,MarcelDekker,1985,

[4]

JerónimoLópez‐Salazar,Metrizabilityof spacesofholomorphic.functions,J. Math.Anal.Appl.355_(2009),no._1,434‐438.

[5]

J.Mujica,Gérmenes holomorfosyfunciones holomorfasenespaciosdeFréchet,Pubhcaciones delDepartamentode Teoria deNnciones,Universidad deSantiago, Spain,1978,

[6]

N.Weaver,Lipschitz Algebras,World ScientificPublishingCo.,Singapore,1999.

DEPARTAMENTODE_{MATEMÁTICAS,}UNIVERSIDADDE_ALMERíA,04120ALMERíA,SPAIN

E‐mail address: ajimenezeual.es