A class of generalized best approximation problems in locally convex linear topological spaces

(1)

Internat. J. Math. & Math. Sci. VOL. 20 NO. 3 (1997) 487-496

487

A CLASS OF GENERALIZED BEST APPROXIMATION PROBLEMS

IN

LOCALLY CONVEX LINEAR TOPOLOGICAL SPACES

HORAKRISHNASAMANTA

DepartmentofMathematics

University ofBurdwan 713 104Burdwan(WestBengal),INDIA

(geceivedSeptember27,1994and in revisedform July 15, 1996)

ABSTRACT. Inthispaperaclassof generalizedbestapproximationproblemsisformulatedinlocally

convexlineartopologicalspacesand issolved, using standardresults oflocallyconvexlineartopological spaces

KEY WORDS AND PHRASES: Best approximation, semi-reflexive, proximal set, continuous

operators.

1991AMSSUBJECTCLASSIFICATION CODES: 49J15,Secondary 49J20, 93C15, 93C20.

1. INTRODUCTION

Thebestapproximation probleminnormedlinearspaceswas consideredby several authors including Burbu [1], Singer[2]. Alsoinlocally convexspacessomeresults were obtainedby Singer[3]. The

principle objective ofthispaperis togeneralize theideaof best approximation probleminlocallyconvex lineartopological space setting andtofindthebest control.

2. STATEMENTOFTIIE PROBLEM

LetEbe a convex subset ofalocallyconvex lineartopological spaceXandX* be theconjugate spaceof all continuous linear functionals defined onX.

Consider theproblem

x

_fX"sup

[(m-x,f) +IE(m)

where x is thegivenelementofXand

IE

is the indicator functionsuch that

[

0 if mE

E

_r(m)

+c

_if

mE

Tosolve thisproblemlet us consider thefollowingdefinition and theorems

DEFINITION1. Anelement E

E

iscalledabest approximationto z

X

from

E

if

sup

I(

z,

f)l

-<

sup

{1(

m,

f)l,

forallm

}.

fx" fx"

(2 1)

ByRemark 1.1([ ],p 174)itcanbe shownthat isthe bestapproximationtoxfrom

E,

ifandonlyif there existsx[ X" subjectto

(2)

where

sup

](m

z,

1)1=

m

e

X.

/(m)

x.

TIOREM1. Anelement 6 Eisthe best approximationtox6Xfrom elements oftheconvex

set

E

ifand onlyifthere exists

x

6X* such that (i) sup

1(5,)1

sup

I(/-

x,

zeXcX’" rex"

(ii)

(x,m- x)

>

sup

I(l- z,f)l

2,

forall m 6

E

where X**isthe conjugate space of X*. PROOF.

Now

wehave

1

sup

I(m-z,f)lg;m

6

X}

f"

(x)

sup

(x0,

rn)-

/x-(x;,x)

+sup

(x;,m)--

sup

I(m,f)12;m

e

X /ex.

(x0,x)

+

sup

I(;,)1

zXcX’"

andtheoptimalitycondition

(2.2)

becomes

1 1

sup

I(/-

x,f)12

+

sup

:x-

x=x"

l(, f)l

(:r,

m

),

Vm E. (2.3)

Inparticular, form l,weobtain

sup

i(e-

,/)1-fX"

2

sup

I(zS,

z)l

o

xXcX*"

which implies condition (i). Consequently from inequality

(2.3)

condition (ii) follows, as claimed.

Conversely,itisclear thatcondition(i)and(ii)imply that isabest approximation, becausewehave

sup

I(/-

x,

f)l

(z,

m

z)

<

sup

I(xS,z)l.

sup

I(m--z,f)

zXcX’* fX"

<

sup

J(l-x,f)l"

sup

J(m-x,f)J,Vm

6

E

yex. /ex.

andso,wemusthave(2.1).

COROLLARY1. If 6

E

is abest approximation ofx6

X

by elements oftheconvexset

E,

then

thefollowingminimaxrelation

sup

I(x

l,

f)i

rain max

(x’,

m

x)

/6X" mE sup

zXcX**

max rain

(z"

m-

z).

sup I(’,x)l--x xEXcX**

(2.4)

holds,wherex"6X*.

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GENERALIZED BEST APPROXIMATION PROBLEMS 4 8 9

existenceofwhich isensured byTheorem 1,isjustthe solution of thedual problem. Thiscompletesthe

proof.

REMARK1. Letd sup

(l(z

rn,

f)l;

mE

E}

be distancebetweenthepointzand the convex

fX" setE.

Then we obtain aweakminimaxrelationby replacing"rain"by"inf’becauseinsucha easeonlythe

dualproblem hassolutions.

Nextwe shall notice several special cases in whichconditions (i) and (ii) of Theorem have a

simplified form. Namely,if

E

is aconvex conewithvertexinthe origin, thencondition(ii)isequivalent

to thefollowing pair ofconditions

(ii’)

(z,m)

<

O,Vm

.

E,

i.e.

z

E

(ii")

(z,

z)

sup

Iz

e; fl

rex"

where

E

isthe polarsetof

K

([4],p. 136).

Hereis theargument. Fromcondition(ii)replacing

z

by

z,

we obtain

(z,z

nrn)

>

sup

I(z

I, f)l

2,

Vrn

E,

Vn N

rex.

because

E

is acone. Thereforewe cannot have

(z,

rn)

>

0forsomeelement m E

E,

that is(ii’)holds. Moreover,from properties(ii)and

(ii’)

itfollowsthat

sup

I(

t, f)!

_< (z;,

z

t)

leX"

_<

sup

I(z,z)l

sup zeXcX’" feX"

sup

I(z

,

I)1

,

/’eX" hence

(z,

z

l)

sup

I(/-

z,

f)l

2.

Thuswe have

reX"

o

>_ (;,t)

(z;,z)

(;,

t)

(z, )

sup

I(z

t,f)l

rex"

and(from (ii)ifrn 0)

(z,z)

_>

sup

I(z- t,/)l

reX"

whichimplies property(ii’). Thereciprocalisobvious. When

E

is a linearspace, condition(ii’)isequivalentto

(z, m)=

0, Vm E

because in this caseE E.

Itshould bementionedthat the bestapproximationbelongsto Ef"l

{z

.X"

sup

<_

d}

and it exists if andonlyif there existseparating hyperplaneswhichmeetE. Moreover,the setof all best approximationsisconvex and coincideswiththeintersectionof theset withany separatinghyperplanes.

When this intersection isnon-empty theseparating hyperplaneisasupportinghyperplaneandisgivenby the equation

(z,m-z)=sup

I(z-l, 1)l

2,

mEX.

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Now,let usstudythe existenceofthebest approximation. Let

Weeasily see that

where

d inf

(

sup

[(m

z’

f)[2

[fx.

rneE _reX" meEo(z;d+)

[

_reX" (2.6)

/x"

THEOREM2.

A

properconvex function

f"

X

oo,

+

oo]

isa lower-semicontinuous on

X

ifand onlyif it islower-semicontinuouswith respect tothe weaktopologyonz.

PROOF.

We

have alreadyseen in Proposition 2.5

([1],

p.

12)

that a convex subsetofalocally

convex lineartopologicalspaceis(strongly)closed ifandonlyif it isclosedinthe corresponding weak topologyonX. Inparticularwemayinferthatepi

f

is(strongly)closed if it isweaklyclosed. This establishes thetheorem.

THEOREM 3. If the convex set

E

is such that there exists an

>

0 for which the set

E

N

S(z;

d

+

)

isweakly compact, thenzhas a best approximationinE.

PROOF. Accordingto relation

(2.6)

it is sufficient torecall thatalower-semicontinuous funon

on acompactsetattains its infimum Inourcase, the functionisobviouslyweakly lower-semicontinuous (seeTheorem2)ontheweakly compactset

E

C

S(m,

d

+

).

COROLLARY 2. In a semireflexive loc,ally convex linear topological space every element

possessesatleast onebestapproximationwith respect toevery closed convexset.

PROOF. Theset

E

CS

(m;

d

+

1)

isconvexclosed and bounded and henceitiswealdy compact byvirtueof the Alaoglu Theorem([5],p. 15).

COROLLARY3. Inalocallyconvex lineartopologicalspaceevery elementpossessesatleastone

best approximationwithrespecttoevery closed, convex andfinite dimensional set.

PROOF. In a finite dimensional space the bounded closed convex setsare compact and hence weaklycompact.

DEFINITION 2. Let

X, Y

be locally linear topological spaces ofthe same nature. A linear

operator

T"

X

-,

Y

_{is continuous if}andonlyifitis bounded.

In

otherwords thereexists

K

>

0such

that

sup

I(Tt, m’)l < Ksup I(t,r)l,

Vt X.

rn"6Y" l’X"

Theset

L(X, Y)

of all linear continuousoperators definedon

X

withvalues inYbecomesalocally

convex lineartopologicalspaceby

sup

.(T,f),

sup{

sup

,(Tl,

m’)l;sup

,(l,l’),<

1}

feL"X,Y) m"(:Y" l"X"

=inf(K;sup

[(Tt,

m’)[<Ksup[(l,l*)[,

VleX}.

(2.7a) I, m’Y" l’eX"

IfY

R,

wefind thatX*

L(X, R),

calledthedual of

X,

islocallyconvex lineartopologicalspace

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GENERALIZEDBESTAPPROXIMATION PROBLEMS 4 91

sup

[(l*, l)[

sup{

ll"

(l);

sup

}.

(2.7b)

lXcX., _l’X"

If

X

isreal locally convexlineartopologicalspace,then

sup

I(z’,Z)l--

supZ’(z);sup

I(Z,Z’)l _<

IXcX’" t, I’X"

THEOREM4. Let

f0

beacontinuous linear functional on a linearsubspace

A

ofalocallyconvex

lineartopological spaceX. Then, there exists a continuouslinearfunctional

f

onthe whole of

X,

i.e.,

f

X*,such that

(i)

f

/A

fo

(ii) sup

I(f,t)l

sup

I(f0,

IXcX’" IXcX"

PROOF. Since

f0

is continuouson

A,

byrelation

(2.8)

we have

fo(m) <

sup

I(f0,Z)l

sup

I(m,t’)l,

Vrn A.

IXcX" I’X"

By

theHahn-Banach Theorem

([

],Theorem 1.10,p.

17)

for

f0

and for the convex function p(x) sup

I(f0,/)l

sup

IXcX’" l’X"

Aspecialization ofthistheorem yieldsawhole class ofexistenceresults. Inthis context weshall present ageneral andclassical theorem concerningthe existenceofcontinuous linearfunetionalswith

importantconsequencein theduality theory of locally convexlineartopologicalspaces.

TItEOREM5. Letmbeanon negative number and let h

B

--,

R

beagiven realftmetion,where Bis anon-emptysetof thelocallyconvex lineartopologicalspace X. Then,hhasacontinuouslinear

extension

f

onall of

X

such that sup

](f,/)l <

rnifand onlyifthefollowing condition holds: IXcX"

EA’h(a)

_<

rnsup Aia/, A,a,l" Vn N, Ai

R, a

B. (2.9)

,=1 lX* 1=1 1=1

PROOF. Fromrelation

(2.7a)

and

(2.7b)

it isclear that condition

(2.9)

isnecessary. To provethe sufficiencywe consider

A

span

B

and wedefine

f0

on

A

by

fo(m)=EA,

h(at),

if m=

A, aaA,

oB.

t=l :=1

First, usingcondition

(2.9)

weobservethat

f0

is welldefined onA. Moreoverfrom condition(2.9)the

con-tinuity of

f0

onAfollows and sup

I(fo, l)l <

m. ThusanyextensiongivenunderTheorem4has all

lXcX.. therequired properties.

THEOREM6. Foranylinearsubspace

A

ofalocally convexlineartopologicalspaceXand

X

there exists

f

X*withthe following properties

i)

//A

0

ii)

f(l)

inf sup

I(/-

m,

l*)12

d(l,

A)

meA_(.I’eX"

iii) sup

I(f,Z)I-

inf sup

I(z

m,l’)l

d(l,A)

IeXcX’" mA I,l*X"

PROOF. Wetake

B=AU{I}

and h B---,Rdefinedby

h(m)

O,rnE

A,

and

h(l)

d2(l,A).

(6)

which isjust inequality(2.9) The desired resultthenfollows by applyingTheorem 5 Indeed,wehave properties (i) and (ii) and sup

[(f,l)l < d(l,A).

Since m

d(l,A).

On the other hand, if we

leXcX’"

considerasequence

{m}

C

A

suchthat

sup[(/+ m,,/*)[

d(l,A)

weobtain

PeA

sup

I(f,l)!

>-

f

Ii,,,Z’)l

sup

I(/+rn, /’)1

lexcx" sup

\l’X" l’eX"

d2(l,A)

d(l,A)

sup

I(

+

m,,/’)1

l’X"

whichimplies sup

I(f,

l)l >_

d(1,

A).

Hence property(iii)alsoholds. IXcX’"

COROLLARY4. Inalocallyconvex lineartopological space Xfor every EXthere exists a

continuouslinearfunctional

f

E

X"

such that (i)

f(1)

sup

(ii) sup

I(f,t)l--IXcX’" l’eX

Moreover,

if

#

0, thereexists g

e X"

suchthat

(i’)

z()

sup I’EX" (ii’) sup

lEXcX’"

PROOF. ByTheorem6,

d(l,A)

sup

I(,l*)l

whereA

(0).

Thenthecorollary completesthe l’X"

proof.

DEFINITION3. Thespace

X

isstrictly convexifevery point ofthepolarset

{

X

sup

l(l,

f)l

l

is an extremepoint.

THEOREM 7. Alocally convex lineartopologicalspace

X

is strictlyconvex ifand onlyifthe followingequivalentproperties hold:

(i)ifsup

I(z

+

y,

f)l

sup

I(z,

f)l

+

sup

I(Y, f)l

and z

#

0thereis

>

0such thaty

fx" fx" fx.

(ii)ifsup

[(z,f)[

sup

[(y,f)[

1andz

-

V,then sup

[(Az

+

(1-A)v,f)[ <

Ifor all)

]0,1[,

fEx"

rex-

fex-(iii) ifsup

[(x, f)[

sup[(y,

f)[

1 and x y,then sup

[((x

+

y),

f)

<

1;

rex" rex" rex"

(iv) thefunction z sup

](z, f)I

,

z X, isstrictly convex.

feX"

PROOF. LetXbestrictly convex andletz,y

X\{O}

besuchthat

sup

I(z

/y,/)t sup

I(x, f)l

+

sup

I(x,

f)l-rex" reX" rex"

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GENERALIZED BEST APPROXIMATION PROBLEMS 4 9 3

continuouslinear functionalz" EX* such that

(x

+

y,x*) sup

](x

+

y,z’)l

EX

Since

(z,z*) <

sup

I(z,

f)l,

(Y,

z)

<

sup

I(Y,

f)l

rex" rex"

we musthave

(x,x’)

sup

I(x, f)l

and(y,

x’)

sup

I(Y, f)l,

i.e.,

fx. rex"

sup

I(x,

f)l’z*

sup

I(Y,

f)l

\rex"

rex.

=1.

Because

X

isstrictlyconvex itfollowsthat

sup

I(, f)l

rex"

hence property(i)holds with

sup

I(/,

f)l’

fX

sup

I(,

f)l

rex"

sup

I(z,

f)l"

leX"

sup

I(z’,)l

.

zXX"

To prove that(i) (ii)we assumebycontradictionthatthere exists z such that

sup

I(z, f)i

sup

I(u, f)l

and sup

I(,Xz

+

(

,X)U, f)l

1, where,X

]0, [.

Thereforewehave

fX fX" fX

sup

I(Xz

+

(1

X)F, f)l

sup

I(,Xa:, f)l

+

sup

I((1

X),

f){.

fX" f.X" I.X

Accordingtoproperty(i)thereexists t

>

0 suchthat,kz

t(1

A)/.

Since sup

I(z,

f)l

sup

I(Y,

f)!

fx" fx"

we obtain ,k-

t(1- )

and so

z----

which is a contradiction. The implications (ii)--,(iii) and

(iv) (ii)are obvious.

Nowweassumethat

X

is notstrictlyconvex Therefore thereexist

z

E

X

andzl,z2

X

with

sup

[(z,z)[

1, sup

[(zl,f)[

sup

[(z2, f)l

1,

zx

a:9_ such that

(xl,z0)

(z2,x)

1,

zXcX’" fX" rEX"

hence

(

(Zl

+

z2),

z)

1. Thus

I(

)1

sup

(Zl

+

z2),

f

sup

(Zl

+

fX"

"

supl(z’,z)l=l zF.XcX**

contradicting property (iii). Hence property (iii)implies the strict

onvxty

ofX.

Now,

from the equality

,sup

I(z, f)l

/

(1 -/)

sup

I(,f)l

)sup

I(z, f)l

/

(1

A)

sup

I(, f)l

fx" fx" fx" fx"

4-

A(1

A)

sup

I(z,/)l-

sup

J(/,

f)l

\fEx" fx.

(8)

494

I

sup

l(Az

+

(1

A),/)l _<

Asup

l(z, f)l

+

(I

A)sup

l(,

fX" fX" fX"

< Asup

[(z,

f)[2

+

(1

A)sup

[(F,

f)[2

fx" fx"

for 1

,

X

th sup

l(z,

f)l

sup

(, f)

d

A

]0,1[.

Ifsup

(,

f)

sup

I(N, f)

weob

fx.

rex.

fx. fx.

thegfi

conve

of thehnion z sup

](z,f)l

,

z

X,

from(ii). Thus the impfifion(fi) (iv) fx.

isestablish d theproofiscompile.

EOM8. If

X

isloclyconvex topolocspace wchisgrimy

conve

thench

elementz

X

possessesatmostone best appromafion th

resp

to aconvex

s E

C

X.

PROOF.

se

by contradion that there

est

two distin best approtions/,/ Since

e

setofbe approtionsisconvex,itfollows that

(l

+

l)

is

so

abest appromation.

i sup

I(m

z,

f)l,

we

Hifd have

mELfex.

0 d sup

I(m

l,

f)[

sup

l(m

l,

fex. fx.

I(

1

=sup

-

(+l),]

Iex"

where

X"

isthe eenjugatespa ofXdereby

In

ew

fthe

cnve

(Theore7)wehave

>sup

(-l)+

(-l),I

sup

-

(+

fX"

wchis acontradion. Tscomples theproof.

2. Ts

prope

iscaefisfic ofthestfilyconvexspaces: if, inalowlyconvex linetopoloc space

X,

e

elementposssesatmost abest approbation th resptto

eve

convext(itis

ou

for

e

seents),

then

X

isfictly convex.

Ind,, if we asse tt

X

is not stripy nveg then

ere

egs z,

X,z

,

th up

]m,f)l

up

I,f)l

P

(m),f)

x-

Fuaheore, sup

I0-),f)l

x,

[o,x].

fx. fx. fx. fx.

Hencethe

orion

hatthe beapprotionth

resp

to

e

cloconvex set

[z, ]

ev

elemt oftsset,dtsclely contradicts the

uqueness.

From

Corofi

2 d Theorem 3 it followsthat in a serefleve strictly loy conv

topoloc space, for

ev

closed convex tgwe c definethenction

PX

X

byPgz l,

where is the bestapprobationof

X

by elemts ofE. Tsnction is ed theprojtion

non

ofthespace

X

intoE. WenotethatPgz gfor

eve

z X.

DEON4. Lusconsider

e

foHog

gener

fyof fionprobls

where

X, Y

elocyconvexlinetopoloc spacesd

F

XxY R. Letus denoteby

(9)

GENERALIZED BEST APPROXIMATION PROBLEMS 495

THEOREM9. Let

X, Y

arelocallyconvexlineartopologicalspacesand

F" X

x

Y

--,

]-oo,

+oo]

be a positively homogeneous and lower-semicontinuous function satisfying the following coercivity

condition

F@,

O) >

0 for any a:

e

XI{O}.

Then, ifepiF is locallycompact, every problem

P

hasanoptimal solutionwheneveritsvalue is finite.

PROOF. Itiseasytoobservethat

H

Projy(epi

F).

By

hypothesis epi his aclosedcone and so(epi

F)o

epiF. Therefore,it issufficienttouseCorollary

1.13 ([1], p. 28) forT Projra and A epiF, taking into account that the separation condition

(1.42)of Corollary1.13([1],p.28)maybe written as condition(2.10).

TBOREM 10. If

E

is aclosed locally compact convex setofastrictly convex locallyconvex

lineartopologicalspace

X,

then theprojectionfunctioniscontinuousonX.

PROOF. Ifa:,-,_{a:, for}_every

>

_{0, then there exists}

n0(e)

_N_{such that} _sup

I(z,

_z,

f)l <

fx.

foralln

> no ().

Denote

whereX*isthe dualspaceofX. Wehave

d,,

<

inf sup

I(a:

m,

f)l

+

sup

I(a:,

z,

f)l

<

d

+

,,

--rnE

[

_f_X" _rex" v,

> ’o()

hence

sup

I(z

egz.,

$)1

-<

sup

I(z.

Pez., f)l

+

sup

I(z=

z,

f)l

<

d,

+

<

d

+

2e.

fx" fzx" fx"

Sincetheset

E

n

S(a:;

d

+

.)

doesnot containanyhalf-line itfollows thatit iscompact where

{

S(x;d+)

y6X; sup (y z,

/) <

d+,

,

>0.

fx.

Thus

i"] S(a:;d

+

)f]E -)

and any subsequence of

PEz,,

has a cluster point which satisfies >0

sup

I(a:-

l,

f)l-

d. Because

X

isstrictly convex, this poimis unique and so

PEa:,,

1

PEa:

as

fX"

claimed.

DEFINITION5. Aset

E

iscalled proximinalifeveryelememofXhasabest approximationinE. That is,theset//7isproximalif theproblem

rain{

sup

I(z-m,f)l}

meE _feX"

has a solutionfor everyx6X.

THEOREM11. Anonemptyset

E

ofalocallyconvex lineartopologicalspace

X

isproximinalif andonlyifepi sup

(., f)[

+

E

x

{ 0}

isclosedinXxR.

(10)

RK. SAMANTA

min{

rex’SUp

[(x-

m,

f)];

mE

’I-

max{(x*,

x)-

PEo{x*);

x* E ,5’"

NE

o

}

foreveryx

X,

where

E

isthepolarsetof

E ([4],

p.

136).

PROOF. TakinginTheorem2.11(Chap.3

[I]),

f

]’E,S sup

I(.,

f)l,A

I,

fX"

weobservethat

episup

l(., f)l

+

E

x

{0}

rex"

asclaimed.

REMARK

3. Itiseasyto seethat epi sup

I(.,f)l

-cone((0,

l)

l),

andsoireisacone, fx.

then epi sup

](., f)]

/

E

0

is closed in

X

R

if and only if

(0;

1)

/

E

is closed in

X.

In fx.

particular, if

E

is a linear subspace, denoting by

0" X

-,

X/E

_the _{canocal mapping, the above}

conditionsaysthat

0E

((0,

1))

isclosedinquotientspace

X/E.

ACKNOWLEDGMENT. The authoris mostgratefultoDr. R.N. Mukherjee of the

Department

of Mathematics, University of Burdwan, Burdwan,

W.B.,

India, for his generous encouragement and suggestionsatvarioussteps duringprepararation ofthispaper.

1]

BARBU,

V. and

PRECUPANU,

TH., ConvexityandOptimizationinBanach

Spaces,

D. Reidel

PublishingCompany (F.st

European

Series),1986.

[2]

SINGER,

I.,

BestApproximation in Normed Linear

Spaces

byElements

of

LinearSubspaces, SpringerVerlag,Berlin, 1971.

[3]

SINGER,

I., Generalizationsof methods of best approximationtoconvexoptimization inlocally

convexspaces,I. Rev. Roum.Math.Pures. Appl.,19(1974),65-77.

[4] YOSIDA,K.,FunctionalAnalysis, Springer Verlag, Berlin,1965.