L1 spaces fail a certain aapproximative property

Internat. J. Math. & Math. Sci. VOL. 21 NO. (1998) 159-164

159

L

SPACES

FAIL A

CERTAIN APPROXIMATIVE PROPERTY

AREF KAMAL

Department

ofMathematics and

Computer

Sciences

U

A.E University

P.O Box

17551

AI-Ain,

UNITED

AKAB EMIRATES

(Received May

4,

1994)

ABSTRACT.

In

this paper the author studies some cases of Banach space that does not have the

property

P1.

He showsthat if

X

glor

.L

(ft)

forsomenon-purelyatomicmeasure#,then

X

doesnot have theproperty

P1.

He

also shows thatif

X

Coo

or

C’(Q)

forsome infinite compact Hausdorffspace

Q,

then

X"

doesnothave theproperty

P.

KEY

WORDS AND

PHRASES: Property

P,

classicalBanachspaces

el,

L (ft)

e,

compactwidth 1991

AMS SUBJECT

CLASSIFICATION CODES:

41A65.

1.

INTRODUCTION

The Banach space

X

is saidtohavetheproperty

P,

ifforeach E

>

0 and eachr

>

0, there is 6

>

0, suchthatfor eachzandtin

X,

thereisz E

B(z,

satisfyingthatfor each 0with0

<

0

<

6

B(x,

r

+

)

B(y,

r

+

O)

C_

B(z,

r

+

O)

where

B(x, r)

istheopen ball ofradiusrandcenteredatz,and

B(x, r)

is itsclouserz

The property

P1

plays

animportant roleinapproximation theory, andmanyauthorsusedit This property appearsinapproximation bycompact operators, simultaneous approximationand other areas

(see

for exampleRoversi

[I], Lau

[2],

Mach

[3]

andKamal

[4]).

Math

[3]

showed thatifX isuniformly

convexthenithas the property

P1

[3],

andthat ifX

U(Q),

or

X

B(Q)

then

X

hasthe property

P1

[4].

Mach

[4,

page

259]

askedifthe

space

L1 (kt)

has the property

P1

In

this

paper

the author studies somecasesofnormedlinearspace

X,

forwhich

X

doesnothave the property

P1

In

section2, it isshown that if

X

71

then

X

doesnot have theproperty P1, and in section 3, it is shownthatif t is anon-purelyatomicmeasure,then

L

(#)

doesnothave the property

P1

Thesetworesultsgiveanegative answerforthe questionof Math

[4] In

section3,it willbe shownalso

that if

X

(goo)’,

or

X

(C’(Q))*,

where

Q

is an infinitecompact

Hausdorff

space,then

X

doesnot

have the property

P1

In

this

paper

e

is the Banach

space

ofall real

sequences

z

{z,}

satisfying that

E

togetherwiththenorm

Ilxll-r

Im,

Also istheBanachspaceof all realn-tuplesx

(x,x2,

...,zn)

together

withthenorm

Ilmll

Im,

I.

t=l

2.

I1

DOES

NOT RAVE TRE PROPERTY

Theproof of the fact that

g

doesnothavethe property

P

dependsonthe behaviorof theproperty

P

in

g

In Lemma

2.3,itwillbe shown thatif

>

0isfixed, and

6n

corresponds

to for

X

e

in

Lemma

21, then

6,

0whenn oo,sousing the factthat

g’

is anorm-one-complemented subspace

LEMMA

2.1. If theBanaehspace

X

has theproperty

P1

then foreache

>

0,there is5

>

0such thatforeach y E

X,

there is z E

B(0,

e)

such that if 0

<

0

<

5then

B(0,1+6+0)

NB(y,I+O)

C_

B(z,

1

PROOF.

Letr 1and lete

>

0 be given

By

the definition of the property

P1

there is 6

>

0 such that for eachxand y in

X,

there is z

B(x, e)

satisfyingthefollowing;foreach 8’such that 0

<

0’

<

6’

B(x,

+

6’)

Fl

B(y,

1

+

8’)

C_

B(z,

1

+

Let

x 0and6

1/2

6’,

thenfor all0satisfying0

<

<

6;

B(0,1

+6+)CB(y,

1

+0)

C_

B(0,1-t-6’)B(y,

1

+0)

C_

B(z,

1

+0).

LEMMA

2.2. Letn

_>

3 be apositive integer,let 6

>

0begivenand let

(zl

z)

beann-tuple of real numbers

If z, :>

6,

and for each

_<

n-1

Z

+...

+

z,_1 z,

+

z,+1

+...

+

z _<

6,

then

[z[ >_

(2n

3)6.

,=1

PROOF. For

each 1,2 ,n, lety,

z

+

+

z,-1 z,

+

z,+

+

+

z,,,then

=1 =1

=

(n-2)

-

..

,=I ,=I

Therefore

I.,I

_> lu,l _> (

2)6

H-

(n

1)6

(2n

3)6.

LEMMA

2.3.

Let

n

>_

3 be a positive integerand let 6

>

0 be a given real number such that

(2n

5)6 <_

1 Then the elementx,

(6,

6 6,

(n

2)6)

in

g

satisfiesthefollowingconditions

(1)

V0 suchthat0

>

0

B(O,

1

+

6

+

O)

0

B(x.,

1

+

O) :fi

.

(2)

Ifz6

g

and for each 0with 0

<

0

<

6;

B(O,

1/5/

O)

n

B(x,,

1-t-

O)

C_

B(z,

1-t-

O),

then

llzll

_> (2,

3)5

’=lifi=

PROOF.

Let

{e,},=

be thestandardbasis in

g’,

that is

e,

(x,

x,

...,x),

where x and x 0 if

:/:

j and let

n-1

(n

2)Seth

(5,

,

5

(r- 2)5) e

e

X 5

ez

=1

Then

Ilxll

(n

1)5

+

(n

2)5

(2n

3)5 <_

1

+

25

<_

2

+

5,

therefore for each 0 such that 0<0<6

B(O,I

+6

+

O)

F’I

B(x,,,

I

+0)

=/=

.

Assume

that z

(zl,

z2,

z)

E

g

is such thatforeach 0 with 0

<

0

<

5

Lx

SPACESFAIL A CERTAIN APPROXIMATIVE PROPERTY 161

It

willbe shownthat

(1)

z

_>

6,

and

(2)

foreach

_<

Z

+...

+

Z_I Z

+

Z,+I

+...

+

Zn 6.

Ifthese aretruethenby Lemma 2.2,

Ilzll

Il

(2n

3)6

=1

(1) Assume

thatZ

+

+

z

<

6.

Let

,-2

1

+

6 1

(2n

5)

,=1 2

=(6,

.,6,

1+6

1-(2n-5))

’

2

Then

1+6

Ilull-

(-

2)6+---

+

1

(2n- 5)6

=1+6.

On

theotherhand

I1

11

1

+

6 6

+

+

(n

2)6

2 2

--1.

Thus,

foreach0 such that 0

<

0

<

6,

y E

B(0,1+6 +O) f3B(x,,l+O).

But

SOforany0<

(6-- =lkZ)

y

"

B(z,

1

+0).

(2) Assume

thatforacertain

io

_<

n 1

2;1+.’.

+

go-1

go

+

Zo+

+...

+

Zn

>

6.

Let

(1+6),

(__)

(

1+6

(1+6))

Y

2

%

en

0,0,...,0,----,0,...,0,

2 E

e.

Then

io-th

term

162 AKAIVIAL

IlY-Zll=(n-2)+

--

+

+(n-2)

1 1

(2n- 5)a

(- )

+

+

2 =1.

Thus, foreach8 such that0

<

8

<

,

y6

B(0,

1

+

+

8)

B(x,,

1

+

).

But

e

l(y

+-..

+

y,o-

,o

+

,o+

+

+

y,)

(=

+

+

=,o-

=,0

+

=,o+

+

+

]-

1

(z

+...

+

z,o_ %

+

Z,o+

+...

+

z,)l

I

+

[

+

(z

+

+

=,o-

=,0

+

=,o.

+

+

>1.

Thus,

forsome8

>

0, y

B(z,

1

+

8).

TEOM

2.4.

l

doesnothave the

prop

P

PROOF.

It llbesho thatforeach

>

0, thereis

xe

6gl,such thatif z6

g

dfor l8 th

0<<gitistmethatB(0,1+g+)B(x,l+)B(z,l+),

then

lz[[ >

.

Let

{e,},

be the stdd basis in

g,

md let

>

0b given If

>

1thenforeach 8

>

0

B(o,

+ +

)

n

B(=,

1

+)

g

B(o,

+

+

)

n

B(=, +).

Thusone can

te

Xl tobex

So

thoutlossof genrgity one may assumethat

Let n 3be apositiveimeger satisng

(2n- 5)

1 d

(2n- 3) >

,

md let x, be as in

Lena

2.3 Define

n--1

=

,

(

=)e.

(,

, ,

(.

=),

o,

o,

...) e

e,.

z=l

Then

[lxel[

[lx,

2

+

,

thus

B(O,I+6+8)B(xe, I+8)#

for 0<8<6.

Let

P,

gl be themappingdefined

by P,

({

x,

},

x,

},

By

the constructionof

x

itsimage

under

P,

is the elmentx, Assumethatforsomez6g

B(O,I++8)B(x,I+8)B(z,I+8)

0

<’8

<

,

thenin

g

s(o,

+

+

)

s(=.,

z +

)

s(P.(z),

z +

)

o

<

<

,

Thus

by

Lena2.3

[lP,(z)l[

(2n- 3)

>

.

Therefore

1

I1=11

IIP,(=)ll

>

.

3.

OR SPACES T DO NOT

E PROPER

P

The

subspacc

Y

of

X

is cled a

no-one-complememed subspacc

of

X

if there is a line

projection

P"

X

Y

tisngthat

[P[[

1. If

A

is asubset of

X,

md x6

X

then

d(=, A)

i{ll=

11;

e

A},

L1SPACES FAILACERTAINAPPROXIMATIVE PROPERTY 163

The compact widthof

A

in

X

is definedby

a(A,

X)

inf{ 6(A, K); K

is acompact subset of

X}.

The compact width is said to be attained if there is a compact subset

K

of

X

satisfying that

a(A,X)

5(A,K)

In

this section it will be shown that if

X

(C(Q))*,

where

Q

is an infinitecompact Hausdorff

space,

X

(o0)’,

or

X

L1

(#)

where # isnon-purely atomic measure,then

X

does not have the property

P1.

Theproof of the following propositionis

elementary

PROPOSITION

3.1. Let

X

be aBanach

space

thathasthe property

P1,

and let

Y

be a closed

subspace of

X

If

Y

is anorm-one-complemented

subspace

of

X,

then

Y

hasthe property

P1

COROLLARY

3.2. If_#is

non-purely

atomicmeasure then

L1

(#)

doesnothave the property

P1

PROOF.

By

Feder

[5,

Theorem

2],

LI[0,1]

has a subset

A

for which the compact width

a(A,

LI[O,

1])

isnotattained,thusbyKamal

[6,

Theorem

4.3]

L[0,1]

doesnothave the property

P,

but by

Lacy [7,

sec

8],

L[0,1]

is a norm-one-complemented subspace of

Lx(),

therefore by Proposition3 1,

L1

(#)

doesnothavetheproperty

P.

NOTE 3.3. Theorem 2.4 togetherwithCorollary 3.2giveanegativeanswertothe question of Mach

[4, page 259].

COROLLARY

3.4. fix

o

or

X

C(Q)

forsomecompactinfiniteHausdorffspace

Q

Then

X"

doesnothave the property

P1

PROOF.

If

X

eo

then

e

is anorm-one-complemented subspaceof

X’,

and if

X

C(Q)

then

byKamal

[8,

Lemma

3.2],

el

is anorm-one-complemented subspaceof

X’,

inbothcasesone concludes

by Proposition3.1that

X"

doesnot havethe property

P.

REFERENCES

[I]

P,

OVERSI, M.,

Best approximation ofbounded functions by continuous functions,

J. Approx.

Theory41

(1984),

135-148

[2]

LAU,

K.,

Approximation bycontinuousvectorvaluedfunctions,StudiaMath.68

(1980),

291-298

[3]

MACH, J., Best

simultaneous approximation of boundedfunctions withvalues incertain Banach

spaces,Math.

Ann.

240

(1979),

I57-164.

[4]

MACH,

J., On

the existence of best simultaneous

approximation, J.

Approx.

Theory25

(1979),

258-265.

[5]

FERDER, M.,

On

certain

subset

of

LI[0,1]

and non-existence of best approximation in some

spacesofoperators,

J.

Approx.

Theory 29

(1980),

170-177.

[6]

KAMAL,

A.,

On

certain diametersof boundedsets,

J. Port.

Math. 51

(1994),

321-333

[7]

LACY,

H.E.,

The

Isometrtc

Theory

of

Classical Banach

Spaces,

Springer-Verlag, 1974

[8]

KAMAL,

A.,

On proximinality andsetsofoperators

III,

Approximation byfiniterank operators on