View of Antiproximinal sets in Banach spaces of continuous functions

t26 Þ. canaMnñ

10 I\IA1'FIEMATICA

_

RPVUÍT DiANALYSE ^NUMÉRIQUÉ

ET DE THEORIE Dtr L'APPROXIMATION

r,'aNALySENUMÉRIQ*'ät-ir:i#ä:litt"ljrlu^pp*oxlMArloN

RE¡'ERENCES

[1 j C a 1' a m R. R. a u, S. S. Române ^p Rornâne,. e t r a, n-d,inrcnsional quøsiconfortnøl 1074.Bucure¡ti t9ôe; _,,ÀLacus È;"*",,, xËít (ecf) rtaþþings. Þdit. an¿ Edit. Acad.¡\cad.

'

ri;:;i;:i",y;;h;,#'ï"#nlÍ'il åli:lr,

^_

uas,iconformal møþþi,ngs in ^sþace. Trans. Ame¡, cient of quøsiconformatity. Acta Math. ll4, I maþþings in thyee sþøce. (pretintinary report).

( I e57).

rresþotr.dence

_of . a q-uasi,aottforntal ntaþþittg in Army. 1.he Univ. of Wjsconsin. uni,,IeSfrrii_

1e63).

n.taþþings

,in

sþace. Ãnn, Acad. gci. ["orr.

ses

of

d,ifferenti.able Junctions. Arkiv för Math.

nd p_caþaci.ty. Ilichigan t,Iath. J. ^{lG, 43_S1}

[ll]

^{3 o p u} AxaÃ

u, B. Havx

(1962)._{oôuo¿o rcìacca} omoôpameuutí s npocmpaHcnee, ficxa.

[12] "On

¿lrcttettmo, nocpeôcmaotrt ceqcttuti.l{otr,r. Axa^ Hayx

CCCP

_).

ANTIPROXIMINAL ^SËTS IN BANACH SPACES OF

CONTINUOUS ^FUNCTIONS

by

s. coBzas

(Cluj-Napoca)

Receiued l.VII.1974.

I.et x

be a normed linear ^space,

LI

attof'.-void. subset

of x and

x e

x.

We

use

the following

notations:

rt(x,

M) : _inf

_{l

_lx - ^{yll: )'}

^e

^M} -

the distance rro'rn x

to

IVI ;

P*(*) : þ

^e

^M:||x - yll:

^d'(x,

^I,l)} - ^the

^metric

plojectiou;

E(IVI)

: _{x

e

X: P*(x) A

^Ø}_

'r'lre set ¡Z is òatled þroxim.inal ^í1 E(

: M

(see

tsl) -

Follor,ving v.

of

all

normed linear ^{spaces r,vhich} set, and

by N,

^{the class}

^{of all}

^norme

oroximinal

bounded closed convex

Lr,

^.u,. ot,rJg

16], a

Banach spacc ^b

is rrou-reflexive.'I'he

chatacterizati

is

rnore complicated. The

first

exam

given by

M. EDDLSTETN

and ¡.

lho."u that

^{the space c} belongs al

tion; all

undefined terms

are

^as

ces

X, Y

are isomorþhic (notation phic bijection g :

X -' ^Y.

^{'l'he map}

^If

further, llqiø)ll : _lløll for

^all

ornorþkisnl, and we say

tlral

the ^spaces ù'c (notatíon'

X:Y)'

I,et

S be a compact Hausdorff spate and C(S) be the.!aga-cþ ^{space of}

all real -

valued, cåntinuous functidns defined òri S.

en

id,eøl

I

^{is a linear}

;;br;;"" ^or õijf't""tt tt'"t

^{xy e}

l r.or all x

e

I

^a''ð'

_v

^e

^{c(s)' rf S' is}

^a

Iza

s. coBzAli

2 3 ÀñriþnoxivrñAl ^sETS t29

sLrbset

of SletZlSr):l*.,C(S)

:ø(s)

:0forall _se S.]. Thel, _1is

a closcd ideal

in c(s) ii

and only

ii

tnerà is a crosed subsef

sr'ir s ,"åi ti"i I :

^Z_{Sr) (see

l8l,'p.

^trO)-

\\/e are now

in

position

to

state

h non-void interior. ^'We asree convex

body. It

follows ãhai

to

a convex cell

M in

a nor_

on X

equivalent

to ll ll,

^and

^M : ll'llr ^{is an}

equivalent riornr

on

_X,

ls a

convex cell

in

X.

rc'ce that i'

^an expansiorl coo,

,, * ïfîlï;:'# j.iìr"îllff,Íl'j:

n,

to. be zero.

'Ihe

occrlrrence of

ã

numbe,

nr: ö, ,r*.", ilrat the

corrcs_

pond_ing

It

is r,vell

te¡m

known misses,

that e.g. c is

<¡3.0 isomo

l-

øtural, numbcr.

If

^euery i,nfi,ni,te

an

øntiþt,oximinal _{conuex- cel,I,}

in

C(aþ

.n)

_contains

_øn

_anti-

Proof'

Let I

be au

infinite

dirrensional closed ideal

i' c(o,h.n). ,rhe'

l-lrcre exists

a

closed set

;\ c

_[1, <oÞ,ø]

such that I : Z(t\). ^p,rt

Är:

_lore.(i

- ^l) I I,

l.it'..r1,

ln: _fl, <or.ø]\À¡

and

Zn: Z(l¡)

Lor

i:7,2, ...,

^n,.

Is

^{O is a set and}

I ^q

O we denote

by l"

the characteristic

function

of the set

l,

^i.e'

ll ^for lel

x.(l) :io

to, leo\t.

'Ihen

Z¡

:

_{x.X6. : x e C(roþ.n)}

=

^C(4,).

Since An is homeomorphic

to

_[1,

.o] it

follows

that

Zn

=

^C(qÊ)

^(i,e.

^they

are

isometrically

isomorphic).

Put Âr: Â lAn ^and Xt: {xeZ,:x(a):

:0 for øe.4,¡), for i:L,2,..',

^n.

then Xn

is a closed id.eal

in Zr,

and

(1) I:Xt@...@X,,,

(the direct sum holds algebrically and topologically)-

For all xeI, lt: xtl...+

^xn,

xieXn, i:7,2,...,tr, we

have

(2) llxll: rnax{llx'.|'','..,llx,,ll}.

We intend

to

show

that without

loosing the generality we can suppose

that all the

ideals

X¡, in the direct sum

(1), are

infinite

dimensional. Indeed,

let

us stlppose

that fl, ^nf : MtU fuIr,

a:nd

X¡ is finite

dimensional for

i

e

M,

and.

infinite

dimensional f.or

i

e

Mr.

The ideal

X,

^is

finite

dimen- sional

if

^and

only if the

set

A¿\Àr is

finite.

PutA' : U (4,\,),

^nx

: card('), M!:{ir, ....,ip}, ù1 ... ^1ip,

ie Mt

Mr: {ir, '.., j,), j, < ^...

anã

tt

e set -ô.,

is

closed,

it

follorvs

that all

the accumulation points of the

set A,

belong to the set

Ä,,

so

that

the sets Än and An are homeomorphic.

Therefore,

there exist the

homeorph.isms:

gi,t Li,n

fnt, -l-

1,

coÀ],

..., qir: 6*^r, _9l,i

¡,

-->4,-t,

.

"',

9,p: l\¡o

n

L','.

Let ü:À' * ll, ml ^{be a} bijection,

and

let us

define

the

^rnap

g: _[1,

^coh'

nf * ll,rah'

^tr'f

ç(ø)

: _- IÙ("ì'. "t1

ì çr (*),

^{o e}

Ar\A',

'i

: _1,2,

. '

.,

n'

Then

g

r,r'ill be

a

homeorphism

of

_[1,

.Þ

'

n] onto

_[1,

.u '

tr'f

,

^{and tìre mäp}

1:C(<oÈ

'n) *C(ah 'n),

defined' by

Tx(æ)

:

x(q(e")), ^ø

e [1, .lh'

^tr'1,

^x

^e

C(t¡h'

n),

will

be an isometric isomorphism of ^{C(<oÞ '} n) onto ^C(<¡'t ' ø). This isomorphism maps the ideal 1 onto an ideal

/ : ^T(I),

^{of the}

^form:

':';:, ^{: ä i, ãì;;.. å zØ:) :}

2 - Mathematicô - Revue d'analyse numêrit¡ue et de théorie de I'approximatlou. Toroe 5, N0 2, 1970.

130

(see

l3l, IV.

6.3) countable, then

(4)

and (5)

S. conZ.r.$

4 5 ÀN IPROXIMINAL ^SETS

íai

where

Y are infinite

dimensional closed ideals

in Zn, i: l, ....,

t.

Therefore, we can suppose

!h?t I ^{is a}

^{closed. ideal}

^{in cia, .¡¡ ol} th" i;r;

I:.Yre _... Z¿,'i:1,2,,...,1. ^ØYt, Yi being infinite

dimensional'closéd

ideals i;

. .Lgt

us-the'supp_ose.that

i'the

expansion

(r), xo

is an

infinite

dimen- sinal closed ideal

in 2o,i :.I,.2, ....,'n.

Sincà 2'o

='C('þ), ¡V frvpotfrlrìì

X¡ will

contain an antiproximinal corlvex cell

Vn,'i : _i, d,,'

_, ø. We rvill

,

^{is an} antiproximinal convex cell in 1.

and topologically,

V is

a convex cell

^:r" i"j'iiï: .'iT ;:"i L;,7

2,....,n, By

(2)

0 < llr -lll:max {llxo-y,ll:i:L,2, ^,..,n}.

l1tt N-l: {?11r,.ry1.:llry.-

^y¿ ll

:

_llø

_-yll} and N,:

_[1,

ø]\N,.

N, fØ,

^and

^for a| i

e N,,,

ltxi.- y;ll : _{l)*'_ lll>} 6, ro'tr,ui tÌ _a

the set

lllx;-yil.rc v,

^being antiproximinal,

ieNrattdxreVr,

_there prit exists

y!,:U" j'o'. _{vr, "e}

_such

Zr. that

Set _llø,

-

y':Dú]-)y,

i-/Vr ¿=JV,

[*""Sr""that I

^yo

:0).

Then,

putting maxØ:0,

we have

llx,,- y'll.: _max

(max{ll x¿

-

^!,¡

ll:l eN.}, ^max _{ll*, ^{_ y)l:} i €Nr}) <

<llx -

^yll.

Therefore,

_By the

set

Z is

antiproxirninal,

e.E.l.

I.emma

3, it

follows

that

we can suppose

(3)

_C(.s)

:

C(.0).

rf ìl^i,s .a compact

r{a,1dorff

space then,

the

conjugate

of the

Banach .p?:g^g(S) _e

_M(S), is the

space

M(S) of

alî

reg'tar

Éoret

tr"å.ír"r on S. F"; p;

rve havc

llrrll

: u(p, S)

(the

total variation of

_¡r)

and

v.@)

:

J tt'l' ^dp(r),

^x e C(S),

(Irr l3l ^the

^space

^{M(S) is}

denoted

by rca

(S)).

If S

is llpll

: ),=rltr(s)l

p(r) : X,-5ø(s)'

_F(s), ø e C(S)

Conversely, every

function p:S --'R,

such

that

)"=51¡.r(s)l

<

^{oo, defines}

through (5)

a

continuous linear

functional on

C(S), whose norm

is

(4).

If

ø

is

an ordinal number, we denote M(æ)

: M(ll,

ø]), i.e. the conjugate of

the

space C(a).

l|he

isomorphism

A

fuom

the following

lemma, was constr-ucted by

l{.

ÞDEr,srErN and. A. c. THoMPsow

l4], in thé particular

case

of the

space co.

'Ihe

proof given here is the same as

in _[4]. If

S is a compact Hausdorff space, we denote

by

^8"

the

evaluation functionals on S, i.e.

(6)

^ò"(r)

:

x(s), x

eC(S), seS.

I,

e m m

a 4. Let S

l¡e

a

^{countable artd.} comþøct Høusd.orff sþøce, S, ø closecl subset of S,

I :

Z(St)

ø

closed. id,eal

in

C(S), S,

: _S\S, *Ø,

^{u, e}

e M(S),

_llø,ll

<

^ø

1I, ^{s e 52,}

^and.

^g

^e

^M(S)

^{definecl by}

B.:Ð"+ør,seS2.

If for

^all,

x e Z(Sr)

the

function Ax

d,efined. by

Ig,@), s ess,

,4r(s):10, _s

e51,

belongs to

Z(S),

then

A is

a.n isomorþh,ism of Z(Sr) onto Z(Sr), ønd, its ad,joint

A*

aerifies the rel.øtions:

z4*à":$", s€52.

Proof

. By the definition

of.

A, it is

clear

lhat A is

a

linear

operator.

Put

D:{x

^e Z(Sr)

:

_lg,@)l

(

1, s e Sr}.

Dvidently, ^{D is}

^a

symmetric,

convex, closed subset oI Z(Sr). Since ll*ll

<

< (l + ^ø)-l

irnplies lg,(z)l

a

^{1, for all}

^{s e}

^52,

^it

^{follows that 0}

^is

^{an interio¡}

point

of

D.If. x ^e

^Z(Sr)

^{is such that}

ll*ll

> ^(l -

^{ø)-1, and}

^x

^e

^{S, is}

^such

that

_/x(s)/

:

lløll, then

le,(ø)l

> lxls)l - lu,(x)l>

llxll

-

^{ø . llxll}

: (l -

^ø)llril

^{> I.}

Therefore,

x 4 D

^anð.

^{D is}

also bounded.

If B

denotes

the

closed

unit

ball

of Z(Sr),

then

D: {* eZ(51):lg"@)i <

^1,

_{s =} Sr}: {x ezls,):lAx(s)1s1, s eSs):{x ez(Sr):llAxll( 1}: ^A-'(B).

Since

D is

a convex cell

it

^follows

that A is

an isornorphism of

Z(S.)

onto

z(5,)' Finally

.,4*\

(ø)

: _t"

(z4ø)

:

Ax(s)

:9,@), x e

Z(51),

s e

52, i.e.

/*8.-gs, Se52,Q.E.D.

Then, +

vi,

),i^lt

s

732

^s,

cóBzA$

ò

If,X is

a

that a functio

onvex subset of X. we s"-,

there

exists

^ \'-suþþorls ^the

^sét

¡z-ï

we

de_note

by ,{tr!,:;/Jí)J ^{'# rw,'.}

We mention

the followi from [4];

Lg,*-^a-5.([!J, prop X be'i

Bønøch sþace,

M

ø

non-uoid., bound,ed, closed.

co X

^qnd.

B tie"

cl.oseil

"ri* ^ioîl

of X.

Then, the set

M

^,is

ønt

and,

only1¡

E(M) as(B) :

_{0i.

bv "p,!Y)

^the ^closed

linear

space spanned

by M.

We have

fr QVI):

:y.-W-\.

^Asequence

^{{xo:k, e} N}inaBanachspaceXis

_{called coìmþtetc}

if

sp({øJ)

: _{x. u}

every

x e x

^can

be

uniquely ìepresented.

i¡ the

forni

",= Ð

&¿x6,where

øoe\,

then {xo} is ^called,

a

^bøsis

for X. A

sequence

to {*} ^if lïiJ:.'"#%

For

complet

. ^rn

^the _^sequel

^{we will}

use several

times the foÍowing

_{remma, whose} simple

proof is

omitted.

fr:;'i;::!y,:,h

urther

if _{xo}

ls (resþ.

a

^bøsis)

_ _ \"t,{tn: i e

N} betheusualbasis

of

coanð.

{g,:ieN} ecl:¡rits

^conju_

gate system.

B. Lyt

l)*i, ,!):2, ^{e^N]}

^bg

^aco -

bíortkogonat sys- such

.tkøl

^{the scl Br.}

: !(

-e ^{co :}

I ^1, ; e"ru), ií

^a

co, ønd.

tet

_ll.ll1 be

tht: lriinleoail¿i' iát o¡

tie"set"B,'.

Txr:e¡,ieN,

extend,s

to øn

ísom.etric i,somorþhism

T of

(co, ll.,,lr)

onto

(ro,

ll.ll).

^The

adjoint

oþerator

of

^'f

,

^uer.ifies

T*81

:¡r, _ie

_N.

ANTIPROXIMINAL SETS 133

Proof.

Since

B, is a

convex celi

in

co,

its

i\,Iinkowski

functional will

be

a norm on

c0, equivalent

to the

usual

norm. By the definition of

the Minkowski

functional, we

have

for all

x e co:

llrilt:inf {),>0:x eÀ8.:inf {À{0:À-1 xeB,}:

: inf

_{À

> 0: lå,(i.-t ^. x)l _< t, i

^€

N} : : _inf

_{À

₂ 0:lhn@)l < r, leN} : -sup{llr,(*)l:ieN}.

Put X : sp ({xt: i . N}), Y : sp (ei

:

i e N)}, and

define

the

linear operator

T: X nY by f(D:r::rø¿x¿):D'i:ron

e'¡,

n eN.

Then, bV Q) ^and taking into

account

that

hn(ù

:0 for i > ^n,

^we

have

for

all

y :DI:,

a¡x, e

X,

ll?yll : llDf:' ø,eill: max{lø.,1

_lo"l}

:

max{l

h,(Ð1, ...,

_lh,(y)l}

:

-

^sup

^{lh,(ùl:

^{a e}

^N} :

_llyll,.

therefore, I is

an isometric isomorphism

of (X, _ll'll.)

onto (Y,

ll'll).

^Since

the

system {ør}

is

complete,

X :

co and

T

extends

to a linear

isometry

of

(co,

ll.llr) into

(ro,

ll.ll).

^Because

^the

^norms

ll'll ^and ll.llt

^are ecluivaleut,

there exist two real numbers

ø,p

) ⁰

^such

^{that, øllrll <}

lløll.

<

Plløll,

%

e

co. Therefore,

llTxll:llxllr>

^{allxll, x e co,}

^{From this} inequality,

easily follows

that the operator 7

has closed range. Then

T(co):T(tù=f6):Y:co.

This shows

that T is an

isometric isomorphism

of

(co,

ll'llt) ^orito

(co, ll.ll).

Finally, by the definition of T

^and

by the biorthogonality of

the systems

("i,

8't) and

{(x;, k)) it

follows,

r*ò1(ø¡)

:

'í(Tx¡)

: ìi(r;) -

^à¿;

:

kn@ì.

The

system {ør} being ^complete

in

co, we have

7*8j : kr, i eN,

_Q.E.D.

R e m ^a r

h. It follows that the

sequence

{ør} frorn I'emma

B,

is

a basis

for

co, equivalent

to the

usual basis

of

co.

Proof of

Theorem 7.

As

was

shown

(see

(3)), it is sufficient to

prove 'Iheorern

1 for

a space C(coÈ).

In

order

to

avoid tedious notations, we suppose å

: 3.

The

proof of the

general case proceeds analogously,

7

134 ^{S. C-ISZAS} _B

I,et us then

suppose

that  is a

closed súbset

of [1, o3], and 1:

: Z( ₎ is

an

infinite

dimensional closed ideal

in C("t). We

have

to

con-

sider several ^cases.

Case

I. l\:Ø,

^i.e

. I : C(.t).

Firstly, we give a

complete biorthogonal system

{(t¡,

"f):1 ^< i <

^<o3}

in

^C(ors)

X

^il4(coa).

I,et us

define

the elements

e¡E eC(cos)

forl(lçco2,by

ANTIPROXIMINAL ^SETS 135

I

Lel ^x

^{e C(to3) and- e}

> 0' Using the continuity of ^{the function}

^ø

on ^<ôs, az ^.

h "rr. ^{.ä^.*1Þl} fi +-1, oi.

^can

^find

sucðessively

the

natural

ä"-¡Ltt

^ho,

^lo,

^mo,

| <

^h'o

"lo,

^Ttxo

1lo'

^such

^that (10) lø(.t) - ^{t4i)l < e, lot i} )

^c¡2

^'åoi

(11) lx(^'' (å-1) +i')- ^x(.l.'' å)l{', ^for o'lo1i {r¡2

^and

h:1,2, '.,,

^/to;

(r2) _lx(^'. (å-1) f co'(l-1) -lnt)-x(az'(tt'-l) +''l)l<'' . lor m¡<m <Lù,

^h

:

1,

2, "', ^ho

^and

^I : l' 2" "'

^lo'

LetNo: [1,

^{Þ0], N1}

: _[1,

Þo]

X

11,

lÀ, Nr: [1' ^Éo]x [1' l]xll'

^mol'

and

let

! :

^x(.J.})'

,",*oÐ" lx(^' ^'h) -

x(a})le,''n

I + D ^Lx(rr.(å -

¹⁾

*

^co'l)

- ^{x(^' '} h)fe-'.6-t¡+o'r *

(È' l) e ¡rt

+ Ð lx(^''

^(,ä-1)

!<'t'(t-l)-lm)-x¡o''(å-l) f c,'l)l'

(h'l¡nle¡¡"

,eo¿,(Þ-l) *o.(r-1) tru.

eoualities (10),

(11)'

(12)'

it is

^easy

i'<

^.¡r,

^{thät is}

llx

-

^yll

^{< e.}

^'l'hcle-

Let tl _-- {* ₌

^C(S)

:lløll.< 1}' Bv

a result'-proved

bv s'r'

zuHovrcKr

1-11.1.

in the

case

of a metri" "o-pu"i ^{S, and'by}

n.H. PHËLPS

[7], ⁱⁿ

jenËr"t,

_{¡r e} S(U)

iI ^and

^onlY

^if

(13) ^s(r-'*)

^O

^{s(v'-) :Ø'}

If the

^comPact

S is

countable, then

s(¡.,)

:

_{s e pr(s)

I

⁰

(14)

s(p+) : seS:¡r(s >o)

s(p-) : sc

^s

<o)

€o''(,{ _l) } o.(/ ll+r,(i) :

7 if i:

<¡z

.

_(1,

-- 1)'l-

ro

. (l -

¹⁾

^j-

^nr,

0 in

^rest.

eô..(Þ-r)

+r.r(i):

for 1 < k, l, m

^.'-¡u¿;

1

if

<o2.(/t-

l) f..(l-

^{1) -l-}

^{I <l}

^(co2.(,f

-

¹⁾

^-l

^a.L

0 in

rest

for 1 (

h.,

l

<-¡'o;

(8)

e.'.¡(i)

:

⁷

if a2, (h-1) l- t (

^r.

a

^{co2' À}

0 irr rcst;

for 1<Æ<co,

e,'(i):1, 1<i(ro3

Let

^define

^the

functionaTs

fte

^IV[(az),

¹ ( I

_=< ô3, by

(e)

f^2.1¡,t)+a.(t ^l) r.rr: ^àr,.1¿ r¡-Fr.{t t)+lil - à.,.11-r)+o.r -

-

^8<¡..À

-

^òr"

i ^{for 1} ( h, l,

^nt,

Ill;

Ír,.(¡- ^t) +..t - Ð^,.Jr.- l) *..r - ^Ð.n.¡ - ^à.",

for 1 <

le,

lla;

.f^r.¡. :$.,.¿-àr.

.f"¡ :

8cu'

where 8, are

the

evaluation functionals ^(6).

Lemma 9. ^The

^system

{(tr,.f,):1 ^< i <

^c,r8}

is ø

C(øB)-comþl,ete

biorthogonal, systeno

in.

C(as)

X

M(<os).

Proof of

Lemma

9. The

biorthogonality

is

obvious.

'I'o

corrplete the

proof of Lemma 9,

we have

to

show

that the

^system {øn

: I _ç i < .t}

is

complete

in

^C(cos),

136

(16)

s. coBzÀg

Now, _lr.e u'ant

to

define the functioT?l.

gr-. M(^"),

such

that

sp({g,:

Å"*uårñ".;'Ìì¡ ttrl : _{{0}. r,et} us ;";.id;"ir,"'å'frí

""'tt, "t*ii,iìt,

(15) oo\):2r,-t(2i.+t), 1<x<<,l; I <È<<¿,

(see

l4l, p.

^55a)

Define

9."(x)

:

r(<u')

* 2-r,: ^(_t¡t2-t. x(a, .i);

g,,.n(x): _x(az

_{. n)}

+ Z-"Li:rl- '-' _(- l);2-t

^. _{x(a2 .}

i) {

n ^z-te+aÐ ( r);2-i

^.

_x(^, ^.

onþ)),

for 1 < k<r;

go'.(Á-r¡16.¿(x): x(az'(/,

- l) I ^a.t) t

^g,,.r(x)

-

^x(ø2.h.)

|

-L -r-

L . ,

^{-(h -r r-r zl -,} I (ì<o

\- _{/_(_1)0}

^¡ .

2-¡

. x(az

.

_(h

_

₁₎

+. ^.

^o,(i)),

for 1 < k, l.io:;

8.'.(¡-r)+o,,tt-t)rtn(x)

: x(a, .

(tr

- 1)+, ^.

^(t

-

¹⁾

i

^nù

I I

^g.,.1¡-tt

+..t

^(x)

-

^x(oz

^.

^(ll

- ^{l) +}

^{cù .}

^{l) +}

1-

2-&''

^r

-+ÐÐ"( _l)i)-r _x(a, . (,Þ-l)-l-.. (l_

1)

+

^o,,,(i)),

for 1 <

^{h., 1,,} m

{a.

Proof of

Lem,øtø 10. We

intend to apply

I,ernma

4. Evidently, gr:

:

п

* ^u,

^and llønll

< 2-t

^(see (16). We have

to

show

that Ax

so defined belongs

to

C(<os).

But, if {cr,} is a

^sequence

in

11, to8] converging

to

a,

then it is a routine verification to

show

that

Eo,,(x) converges

to

^go(x),

that

is

Ax

is a continuous

function.

Therefore, I,emma

4

^applies, _Q.E.D.

Put

(20) where

c(.').

(21)

Put

l'22) and (23)

ANTIPROXIMINAL SETS 737

V:{xeC(co3) :lf¿@)l <1, 1<i<co8},

.f, ^are

^defined

bv

^(9).

It

easy

to

see

tlnat V is a

convex cell

in

There exists

an

isomorphism, say

H:

C(oB)

-

^cu.

vt: A-t(v),

10 1l

'l'aking into account the formulae (r4), (rs),

(16),

an

examinatio'

of all

¡rossible combinations

g :

"rg;+

^o;gr:,

^{of iú"-} "íé-"o ts

g¿, shows

that

S(g+)

f\ ^S(g-) :Ø. u

^¡o'

(17)

_Lr

: _{x

e C(coB)

:lløil < t},

(lB) Y:sp({g;:t _<¿ (.'}).

(le) Ihen yns(u) _:{o}.

I,

e m rn

a 10.

7 he oþerøtor A d,ef,ined,by

Ax(í) :

g¿Ø),

L < i <

^{ios, x}

eG(.r), is

an isomorþhism of C(tl.s) onto C(.i.s). Its ad.joint,A* _aerifies

A*}n:g¡, l<i(r,ra.

B,: ^H(Vt) :

HA-L(V),

where ,4 is the isomorphism

of

C(cos) onto C(cos), constructed

in

Lemma ^10.

The

maps

A ard fI

being isomorphisms,

Bt will be a

convex cell

in

^co.

Let

(24) !¡:

^A-Le¡,

(25)

^Ø¡

:

A*.f _¿,

and

(26) x,:

^Hy¡,

(27) k,:

(H*)-1u0,

forl<¿<ro3.

Applying twicc Lemnra 7, it follows that _{(*t,

ht)

: 1 < i ( .t)

is a

co-complete

biorthogonal systen. By

(23), (20), (27) and

the

fact

that (H-r¡*: ^(I1x¡-t

^(see

_[3]. VL

^3.7), one ^gets

Bt: HA-|(V) : _{x e

^co:

(AH-r)x e 4 : {x e

^co:

lf,(AH-t(x)l<

^1,

1<i <.ot): {xeco:l(H-r¡*¿*f|x)l ^{< 1, 1} <i <,,¡'}:

:{xeco:1h,,(x)l < ^1, 1<i <r¡'}.

I,et o:

N

*

_[1,

.t], be a bijection

and.

let us

define

(28)

%¡

:

fro(il

h¡

:

koli¡

Put

138 s. coBzAg

t2 ₁₃ ÀNTIPROXIMINAL SETS 139

for all I

e

N. il

fo-llows-

that, the

co-complete

biorthogonal

systen

{*,, ^hr):ieN}, verifies.ttre

tryfottresis

"or

r,ãmma

g. ö"rrotins bv

_ll.ll,

the

Minkowski

furctional of thé'convex _ceil å, _i¡l-.lrf *ili

be

a norm

on co equivalent

to the

usual no¡m),

it.follows thai

düeïË exists

an

isometric isomorphism

T:

(co,

ll.ll,) _, þí,'ll.ll),

^such

^that

TÌ, :

s;

T*8í :

h.

for,i

e N. Here {¿j} d"notes

the

usual basis

of

co and.

{ò;}, its

conjugate system.

I,et us

define

a

new

norm

_ll

,lb on

co, by llxll,

: _llH-rxll,

,(

e

^co,

wherc

17

is the

sonror¡rhism

(2r). k

^folrows

that, _ll.ll¿ wilr

be

a

norm

o\.r0, cqui'alent to the

usual

ròrm

anð.,

H will

^ÈË

ärr"iso-"tric

isomor_

phism

of

^(C(co3),

ll.ll) onto

^(ro, ll.llr).

r,

e m m

a 17. The set Bf is an

antiþroxirn.inør _colraex

ceil in

^(co,

ll ^. ll,).

Proof of Lem,mø.11.

I.etB - {xe

^co,:llxll

( l}and Br: _{xe

co:ljøllrç

< ^1]. By the definition of the nolm

_ll.¡¡r,

(30) B,: H(u),

rvhere

u

denotes the closed

unit balli

r (c(os), ll.ll). since

I is an

isometric isornorphism

of

(co,

ll.llr) onto

^(ro, ll.¡¡¡,

tt íófio#, til;t ^-

(31) _a :

T(Br).

I,et Y

be defined

by

(18) and

let

(32) Z:sp(Bi: leN)).

By (9), I,enma

10

and

(25),

(33) _{Y:sp({u,: øeN}).}

We intend to apply I,emma

S,

It is well known and

easy

to

see,

that

(34) _s@) : z.

By

l,emma

6, (31), (84),

(32), (29). (28), (27),

and

⁽³³⁾

E(81)

: r'¡E(B) : T*(z) :

sp({E,:

i

^e N})

:

sp (iZ,

: 1 < i < .r]) : (¡/*)-1(y).

On

the

other hand,

by

I,emma

6 and

(80).

3(8,) : (¡/*)-r(E(u)).

then, by

⁽¹⁹⁾

-'

"j',Î', li'J

" :iJrïp,iärÏ lii]

:

B¡,

I,emrna

5, the set B, is

antiproximinal

-tt

^{(ro, ]l'.1Þ), Q.E'D'}

-' _Ño*,

_sinôe

Il is arr isometiic

isomorphism

of

^(C(coa),

_ll'l)-

onto.(co, ll.ll"),

the

^set

tr/r:

I/-1(Br)wi11 be an

antipioximi,al

convex cell

in C(tt), .i'Ëiôt

_{- --} concludei

the proof of

Theorem

1 in

^Case

^I.

nrmørþ..

ny (2gf and the fact that T is an

isomorphi..ol.

of

C(ors)

onto

^c6,

it ^totlówi tûat {xr:i

^e

^{N} is, in fact, a}

^basis

for

^{C(<oB) equiva-}

lent to the

usual basis

of

^co'

'

_Case

II. I : Zl _), ^L

ø-cl,osed, subset

o/ _[1,

^{co3] and' as e}

l\'

The proof is the

same as

in

case

I, with

some changes

in the

^d-efi-

nitions of the

elemente ^e¿, f¿, ^8¿.

since c(<,r3)

is

isomorphic

to

^co, _-and ^co

is

isomorphic _to c,

it

follolvs

that

^C(<os)

ii

iéomorphic

io c.

Thið isomorphisrn carries-.the

ideal 1

onto

á" i"iiàitä

áim"trsioåal closed ideal

in c.

P¡ut, every

infinite

dimensional

"ts"li¿"tt

^{in c is} isornorphic to co. Therefore,

there

exists

an

isomorphism

(35) H:I'co,

sp({s'})

";¿i": äi

(the

analog

of

(19)).

We

observe

that, iÎ f ^{e M(^t) is}

^{such that}

^f(a) : :0 for aeÂ,

then

ll.fll

:D"=ol/(")l : _ll/l'll,

where A

:

_[1,

cot]\^

^and

_f l¡

^denotes

^the

restriction ^of

f ^to L

'Iherefore,

/ ^{is a} ttot--pt"sétrrìttg

"*i"nsion of .fl, to ^{C(.t). We shall}

^{define g, e}

'e M(us),

suchthat g,(i) :0 for

a e

Ä

and

for

g e s/({gr}).

s(g*)

_O

s(s-) +ø'

Then, by the

above

quoted result of R, R.

Ph.elps,

the restriction

of

g to I

does

not attain its

supremurn

on

U'

I,et A : _[1, <o3]\Â

and

let

d1

1ct2 <,..'

be the accumulation points of the set

A of the

form (2e)

r40

and

I,et

now

S, COBZAS

l4

15

Put

ANTIPROXIMINAI SETS 1.47

ct¡:

¡¡2

.

_),,,

1 <

À, <,<,1.

I!

a, e _{_A}

then, by the

closedness

of the

set

 there

exists

a

riurnber

tu, i 1jo l

<¡, such

that

(36) l"' ^'

^(Ào

-

¹⁾

⁺ ^ ^{' lo ]-1,} 02 tol -

^A.

By the

properties

of ordinal

numbers,

it will exist a

homeomorphisrn

(37) _I¿: [,

^{<^,']}

* [.r ^.

^(Àe

-

¹⁾

^{+ a}

^{. te}

-ll,

o2

.

_to],

(rlocanbedefinede.g.by nn(i):c¿r.(À¿-

¹⁾

₊

^.toi_i,forl ( ^i(<¡2,

no(¡.i.'):

^o2

'

^ì¿)

Put

E"o@):

^x(o-h)

+ ^2-tr+tr. Ð (-t¡t2-t.x(r¡[email protected].)),

I d. 1-

-

f.or

i en¿(il,

cor])

in

rest,

for h:1,2,

IT.a.2.

^eh¡

eA O,o''

^(Ào

-

¹⁾

* t'

^{lu, ø2 . ),,01.}

By

(37) there exists

a

number

7'

^such

tt'at ae,¡:r¡*(co . j'). Put 8"u,¡Ø):

^x(an,¡)

l2-ta+¡ D;:î''tl (-l)t2-tx(\o(.'i))) *

-L

9-&+/+tl \\

1 <i<o

(-t|t 2-;*(rlo(.

' oj,(i))),

,.\ f I

f.or

i er¡u

",0,r(r)

:i

o in

rest,

Íon,j:8oo,i-8o'

([1, .'])

II.å,

^øo

4

^A

edoU')

{å

öoo'

g*0,,@)

:

x(oe,¡)

I ²

^{to r'i.r-l)}

rD. (- l)i

2-tx(r¡¡,¡(i)), e,o,¡(i)

:

^Lot

i :

un,i

in

rest

8o¿,

i I

crp]1ctp2{.,,

_, 0

be

the

accumulation

points of the

set

A of the

form dh¡

:ot'

(À*,;

-

¹⁾

+ a,

FLp,i,

I (

À¡,¡, _Va,i

{a,

belonging _ø,,

onlv _is a finite _{the last} .to

numbeJ

^the _of interval

_ør).

gf _fn oo,'.iã'"o{side} _tLis

_Loo-,r,

_ool;

(W-e alsä

put the interval

do

: I

and,

ii,, if ^.;j-irrãià there

are case

there exist th"

hom"ãmorphisms

(38)

'r¡n,i:

fl,

^<ol

* _la¿,j-t;

dn,¡f.

we

have

now to

consider

sorre different situatio's. The

symbor oo wilr have

the

same meaning as

in

(1S).

IL a.

^uo e L,.

Preserving the notations

fro'r

(36),

we

consider trre sub-cases

II. a.l.

^{a.e,i e A,}

f)

loo_r,

., ^. (t, _

₁₎

₊

^ ^.

^tol.

Put

9"0,,@)

:

x(qn,¡)

+ 2-&+i+t),à.(_l)t2-tx(r¡¡,¡

_(i,)),

"no,¡(i): f I for ie _lr'¡([1'

co])

ì

O irr

rest, f _s

J a¡,¡ - ^oep,¡t

where r1a,¡

is the

homeomorphism _(Bg).

1

r-

ah,i

Let now

^pass

to the isolated roints of ll, .tj ^which

^belons

^to

^A.

By_the

homeomorphism

(38),

every isolated

point-ø from A n] ^&*,j

^t,

ø¿,i[

is of

the

form

a

:

_T*,¡(l)

_for a

number

I _{e [],} .[. We

considãt lon,

the following

cases:

ILø.1.a.

øre L,

a¡,¡ e

LO [oo-t,,o' .

^(À¿

- ^{t) +} ^

^.

li

^{(see (36)).}

Put

sn@)

:

x(a)

{ 2 (A+r+1)D::i'tl (-t), 2-t

. x(\r,,(í.))

|

x(r¡o,uþ¡þ)))'

| 2-ln+i+r+t)

1<¡l<o

D (-t¡t .2-t

^'

1 ^for i:u 0 in

rest

-òo e"(i)

:

Jdf ÍLø.2.a..

æ¡

e À,

^a-¡,¡e L,

ñ].t ^'

^(Ào

-_ 1) l. ^.lu,

ocol(see (36))

In this

case,

by

^(37),

there exist l, j' ₌ ll,

^{cof, such}

^that

ø:I¿(<¡.(j'-

¹⁾

₊₄

ÀNuÞnoxtvtNAL ^sETS 143

I+z Put

s, coBZA$

g"(x)

: x(") +

E"u,o@)

-

^x(oo,¡)

I

l6

Å:8,-8o¡,j-8a¡.

II.b.a.

a.p

ø L,

^{ø.¡,7 € À.}

ßr'

(38), there exists

I e ll,

^{co[, such}

^that a:7¡,¡(l). Pú g"(x):

r(o.) -l-

2 ^fttitrt [i-i

t-

^-

ofle can write

Z(It):Xrq OXÀ"OX' in

Z(Tn)

=

^C(o'?),

of

Lemma

3,

^we

ifcoz'ieJ\¿w a

manner' we ^o

or. ZQr).

Z(L):Zr@.'.@Zp,

whereeachZisisometricallyisomorphictoaninfinited'imensionalclosed

iå;;i"t(^1i1" -ci.'¡

such

ihat ^{Ltn-:Ø or}

^{<oÞ G ¡'}

he{1,2,3}' ^{For ø eZ('tt),} x:zt+ "'lzp' ^zi =Z¡''

'll, "',

^{llzrll)' BY. Case}

I ^or

^Case

II

uä t"tipiótäminal

^convex

cell'

^Rea-

L"*-u ^3,

^{one can show Llnat}

^Z(L)

:tt'

REËDRENCÞ

^S

lll

^{A rn i}

r,

D., cou.linuouß fu.nclion ^sþøces. uùh^lltc bott'tttlrtl exlension þroþerly' Bull l{es L^r '^

"---'cãtt".

oI Israel, 10F 133-138, ^(1962).

unaiouote JnHotrcecmsa o npocmpaHclnlax ^{co t7 c}

en "ipàott, 1967.

Archiv ^der Math' 10, 162-169

I,

Warszarva ^1971' zawa 1965.

Berlin-Heiclelberg- New Yrok' Ill|3yxoeuuxuirC.I4.,o¡tuuu¡raltt,llølÍpacluupl:Htnx,lrLneúuorsttþgunt,uouaaoaBllpocmpatclnse

HenpepbtsHbtx,par-i"ñi'I"l';;;;' ft^v;--Ccdlj' ctp n'artn''zt;

s

1to0 - nzz¡'

ncceived 30.6.197G. Utliuersily Babe-ç-,Bolyai' Cl'uj-Naþoca

Instilute of Mathematios'

-f 2-tt

⁺

-i+,r-r)

1<i<o

D (-

^1)o

e,(i):

2-n '*(\o@'(j'-

¹⁾

₊

",(t))),

1 for

tt.

:

a-

0 in

rest

(-l)'

^.

2-¡ ^.

x('r¡x,¡(t))

+

+2-&+i+t+.1) D t_\n .z-t ^.

x(7¡,¡(o,(i))), 1<i<o

I I lori:q.

""(t):10 ^inrest Å:8"'

rr.ð.p

a"h

ê L,

au¡

ê

^L,

If

^q.

:'r¡n,¡(l), prt

g"(x)

- ^x(") i

^z-th)i+t)

D

^f

-l)t ^{.2-1 .}

^{x(^nn,¡.} (",(l))),

l<i<o

eo(i): 1 ^for i:ø.

0 in

rest,

'å:òo

I'his

finishes

the

definitions

of the

elements e¡, -f¡,

gi in

Case

II.

Cqse

IIL I : ^Z(L), L + Ø

ønd' c¡3

ê ^I\.

This

case reduces

to

Case

I or to

^Case

II.

Since

r\ is

closed, fronr

<,f

e ,\. it

follorvs

the

^existence

of a

åo

e [,

^<o

[,

^such

that

fc^r2

'

^(Ào

*

* ^l), .t] ^c A,

where

A : _[1, <ot]\^.

Denoting

A,

:

_ftoz

' (i - ^l) + ^t,

^@'

^' i], f¡ : [, .']\A¡,

¡ :

A¡

fl.t\, X¡:

^Z(/\¡),

f:[1,^r.kol, X:Z(l),