LI(,i) into - CiteSeerX

2 downloads 0 Views 426KB Size Report
Apr 26, 1985 - The functional is the Gateaux derivative of the norm at x e X. Mazur, [4], has shown that the following are equivalent: (i) x is asmooth point of. X.
Internat. J. Math. & Math. Sci. Vol. 8 No. 3 (1985) 433-439

433

ESSENTIAL SUPREMUM NORM DIFFERENTIABILITY I. E. LEONARD Department of Mathematical Sciences Northern Illinois University DeKalb, IL 60115

K. F. TAYLOR Department of Mathematics University of Saskatchewan Saskatoon, SK STN OWO (Received April 26, 1985)

ABSTRACT.

The points of Gateaux and

obtained, where

space.

(,Z,)

Frchet

X

LI(,i)

KEY WORDS AND PHRASES.

into

are

is a real Banach

An application of these results is given to the space

all bounded linear operators from

L(,X)

differentiability in

is a finite measure space and

B(LI(,X)

of

X

Banach spaces, measurable functions, essentially bounded

functions, vector-val ued functions, essential supremum norm, smooth points. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. Secondary:

i.

46B20, 46E40

Primary:

46A32, 28B05.

INTRODUCTION.

Let

m

be the restriction of Lebesgue measure to

[0,i]

and

L=(m,0

the

Banach space of all measurable, essentially bounded, real-valued functions on

[0,i]

equipped with the norm

llfll

In [4], Mazur proved that given any g

L(m,0

{If(t) l:

ess sup

identifying functions that agree a.e. on

t

e [0,I]}

(as usual,

[0,i]) f e

L(m,RR)

f # 0

there exists a

such that lira

does not exist.

jjf + gjj

ijfjj

In other words, the closed unit ball in

L(m, 0

if and only if

V

0

and

The measure V

in

Z

and

We will write

for the (essentially unique) decomposition of e Z with Q O c c Since 0 is a finite masure, there into its non-atomic and purely atomic parts.

d

9d

{A i e I} of atoms i is an atom for V and

exists an at st countable pairwise disjoint collection

for

such that

f e L (v,X)

then

f

the essential value of

is constant

f

on

A

We note that if

A.

d

A

a.e. on

A

and this constant is called

is

ESSENTIAL SUPREMUM NORM DIFFERENTIABILITY If

XI,

X

X2

(El

are Banach paces, and i n is the product space X X

Xn) p

D

I

then the

p

X

2

435

Xn

2

product

p

equipped with the

no rm

ll(Xl I

for

x

(!IXlll

p

p

llx211

+

p

llXnllP) 1/p

+

+

and

p

[[(x I, for

Xn)

2

x

xn)lloo

2

max

llXnl I)

(llXlll,llx21

p

We will need the following lemmas in the discussion of the smooth points of

L,(, X) LEMMA 2.1:

Jo-

llxj 0

>

Xn )

(X

llxjll

for

2

j

Jo

4

is

Xn

Jo

X. 0

(d’ ld’ d

respectively

(x I, x 2,

if and only if there exists a

and

30

LEMMA 2.2:

to

are Banach spaces, then

is a mooth point of

x.

(ii)

Xn

2

X

1 such that

n

(i)

If

X I, X

If

smooth point of

(c, Zc’ c

a finite measure space.

are the purely atomic and non-atomic measure spaces,

in the decomposition of

(L=(d,X)

(,Z,)

be a Banach space, and

X

Let and

L (,X)

fl" then

is isometrically isomorphic

L(U c, x))

The proof of the second lemma is routine, while the proof of the first lemma

(X

uses the fact that

(X @ X 1 2

1

Xn)

X

2

is isometrically isomorphic to

see [3] E) 1 The next two lemmas are straightforward generalizations of results in Kothe [2],

Xn)

we sketch the proof of the first lemma.

LEMMA 2.3:

Let

X

be a Banach space and let

bounded sequences in

X

with the supremum norm.

Lhen n

o

x

is a smooth point of

=o(X)

E(X) If

x

denote the space of

{Xn }n>l

(X)

x

# 0

f and only if there exists a positive integer

such that

llxn0ll

(i) (ii)

x n

> up

{llXnl

n

is a smooth point

@

n

O}

and

of X

O

PROOF. Let

x

(x

n

n_l

(X)

If there exists a mubsequence

be a smooth point

{xnk}k>I__

the existence of distinct elements of

we may assume that

such that

I

we can

llXnl

sup

n>!

demonstrate

k-

E(X)

which support the unit ball at

the following modification of the argument given in

We consider the disjoint sequences

lira llxnkll--

llxll

{x

n

2j j>_l

Kthe [2] and

{x

for

x

(0

n2j-i j_>l

For each

by

i

436

j

I. E. LEONARD AND K. F. TAYLOR

_>

let

i

llxn 2j I

J( xn2j Define

]= j[, (x)

(X) li

Xn

[Xn0{

[x

i

us,

[x[


l

x

e

m

0

(X)

n

0

Xmo

# n 0 with

then

such that

][x

let

(0 (Xm0) x Now #(Yn0) d V(y) (Ym0) for y {Yn}nl e (X) define

x

(X)

at

x

is a smooth point of

A similar argent shows that (ll) must hold as well.

{Xn}n>l y # 0

e

(X)

we have

(x- sup{ m

but

is a smooth point of

is a smooth point of

We have established that if

x

(X)

[[

satisfying

x

and



and

i

and

i

then (1) must hold.

Conversely, If

{Yn}nl

{+j }jl

are distinct support functionals to the ball in

Again, a contradiction.

(X)

and

and therefore there must exist a positive integer

(X)

j (Yn2 j-l)

’’j (y)

and

’. e 9 (X)

This contradicts the fact that

with

with

i

z

If there exists another integer

1’ e X

y

and

j (Yn2 j

j (y)

by

we have sho that if

x]

then

(X)

w*-accumulation points of the sequences

be

and

lljll lljil

respectively, then by construction we have

j}jl

n,

(X)

for.

Let

(x)

e

{Yn}n>l

,

on

?.3

such that

llXnmj_lli

’bJ (xn2j-i

and

X

be elements of

Sj

and

.j

y

for all

{’

and

j

x + Zl[- [

which exists by (ll); them,

(1) and (i) hold, then for any

and

x + %y Xn0 + %Yn0 n

n0})

#

im

A e

Therefore,

IXn0

Yn 0

Is a smooth point of

x

for all

Xn 0

m(X)

This completes the proof

of the lena. #m argent similar to the above gives the following:

LEMMA 2.4: and let

3.

X

Let

(,Z,)

be a finite measure space which is purely non-atomlc,

be a real Banach space, then

L(,X)

has no smooth points.

MAIN RESULT In this section, we characterize the smooth points of the space THEOREM 3.1:

f e

L(,X)

with

f

there exists an atom

(i) (ii)

llfli x

0

A 0

(,Z,U)

Let

@

0

A 0

ess sup

then for

be a finite measure space,

f

is a smooth point of

X

L(,X)

a Banach space, and

Lm(,X)

if and only if

such that

{ilf(o)]l

is a mooth point of

e

A

X

where

O}

and x

0

is the essential value of

f

on

ESSENTIAL SUPREMUM NORM DIFFERENTIABILITY

437

PROOF.

Suppose

f

implies that

Z

e L(v,X)

r

I"

2

llfl fl c

o.

ease 1

atoms for

X.

A

L(v,X)

imply that either

since

rid

L

(Vc’ x) and

L(d,X) (Q ,Z c

e I}

i

m(Ud,X) n

I

while if

(X)

is a smooth point of

up

A 0 Conversely, suppose that

d let

n

>

(i) we have g

and

c

f 8

X

I

lf[ f(0) If L(,X)

e

is finite or cotably infinite.

A

whenever

0

with

ess sup

f(mo

x

0

{[f()[ then


__ [[f[i

(d,X)

with

The re fo re,

a.e. on

L

with

is the essential value of

0

AO

m0

[[f(m0)l[ + ll A

for

and hence

0

r

A 0

and there exists an atom

0

A

l[f(m) + Xg(m)l

(XIX..)

=d

x

[[f(0)-

with

on

a.e. on

A0

Let

6

Let

is a

d

is countably infinite, then

where

L(,X)

hold.

(ii)

and

(i)

a.e.

whenever

f]

In either case, it is easily seen (from

f(>

f x

such that

ib a finite on-atomic

c

is isometrically isomorphic to

Lemma 2.1 or Lena 2.3) that there exists an atom

Z

,

is a paiise disjoint collection of

is finite, then either

l, 2

for

0 on

c

{l[f(m)l[

ess sup

{A.

is isometrically isomorphic to

(ii)

then Lemma 2.1

and

=zc

l]fl?

where

i

i finite, then

X

Since, by Lemma 2.2,

L(d,X)

U

d

be the

fc U rid

fl

L(V d X))o

i ru!ed out by Lena 2.4, since

Therefore,

then Lemma 2.4

(Vc’ X)

6

llf(>ll

is a .mooth point of

smooth point of

Let

{llf()ll

ess up

measure space.

I

(L

i a smooth point of

is a mooth point of

fld Now,

f

> es sup

l[flzdll

Let

V

into its non-atomic and purely atomic parts.

is isometrically isomorphic to

and the fact that

Lo=(v,X)

is a smooth point of

contains at least one atom for

decomposition of

Loo(v,X)

0

f

This implies that

e

On the other hand,

[[ [[g][

>

]l f(0 )[[

A

0}

.

[[f(0)[[-

A 0

f

for then from

0}

A

> 0

I. E. LEONARD AND K. F. TAYLOR

438

"flaee fore,

lit + gil wnr

O
(’flA0 IIT(XA0) I

T(XAO)

(ii)

Radon-Nikodm

T $ 0

(fl,E,)

439

is a point of Gateaux

(Frchet)

and

differentiability of the norm of

REFERENCES

i.

DIESTEL, J., and J.J. UHI,, Jr., Vector Measures, Mathematical Surveys, Vol. 15, American Mathematical Society, 1977.

2.

KTHE,

G., T0Pologische linear Rum I, Die Grundlehren der Math. Wissenschaften, Band 107, Springer-Verlag, Berlin, 1966. English translation: Die Grundlehren der Math. Wissenchaften, Band 159, Springer-Verlag, New York, 1969.

3.

LEONARD, I.E., and K.F. TAYLOR, Supremum Norm Differentiability, International J. Math. & Math. Sci., Vol. 6, No. 4(1983), 705-713.

4.

MAZUR, S., "ber konvexe Mengen in linearen normierten Rumen, Studia Math. 4 (1933), 70-84.

X