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,<)
has no smooth
points.
In this note, we show that an analogous result holds for
of
L (,X)
the space
;J-measurable, essentially bounded functions, whose values lie in a Banach space
X- provided that the underlying measure space
(,Z,)
is non-atomic.
a complete description of the smooth points of
L=(,X)
L(,X)
in the general case.
We show, in fact, that
f
is a smooth point of
achieves its norm on a unique atom for
this atom is a smooth point of
X
if and only if
We then obtain
f
and its (-a.e. constant) value on
I.E. LEONARD AND K. F. TAYLOR
434
An application of this result is
to the space of all bounded linear
given
X
into a Banach space
LI(,0
rom
operators
X
when
has the
Randon-Nikodym
property with respect to
2
PRELIMINARIES
X
Throughout this note,
x
{0}
X
IIIi
1
at non-zero
llxll
(x)
e X
x
X
point of
is a smooth
such that
h e X
for all
x e X
X
,hll
ilxll
point
with
is Gateaux differentiable
such that
O(h)
(*)
0
is the Gateaux derivative of the norm at
The functional
Mazur, [4], has shown that the following are equivalent:
(i) x is a smooth point of X
(ii)
lira
(iii)
llx + %hll- llxll
exists for all
X
the norm function on
X
The norm function on
e X*
if there exists a
Frchet
Of course,
is
e X
h
is Gateaux differentiable at
Frchet
x
x e X
differentiable, at a non-zero
such that
I11
-
X*
if there exists a
liml[lx +
e X
if there exists a unique
The norm function on
A
X*
denotes a real Banach space with dual
x
+
h!l- iixll-
*(h)
(**
0
differentiability at a point implies Gateaux differentiability
at the point.
(Q,Z,)
Let
X
f:
A mapping
denote a finite measure space.
is called
measurable (or strongly measurable) if
(i)
f
(ii)
-i
(V) e Z
V
for each open set
=X
and
N e Z
is essentially separably valued; that is, there exists a set
f
(N)
with
The Lebesgue-Bochner function space
-measurable, essentially bounded,
=
H
and a co,table set
0
L=(,X)
X
such that
=H
f( N)
is the real vector space of all
L=(,X)
X-valued fctions defined on
a real Banach space when equipped with the norm
f
ess sup
{[f()[[:
}
(as usual, identifying functions which agree
v- a.e.)
A set
for any
A e Z
B e Z
is an atom for the measure
with
space
(9,Z,)
rely
atomic if
=
B
A
either
v(B)
or
v(B)
(A)
is called non-atomic if there are no atoms for
can be expressed as a ion of atoms for
Q
v(A) > 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[
<
.,
then
i
j
P e
+
1
by
P
and
#(y)
[
{Xn}n>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
<
[[g[[-
8
0
0
0
I%1
{[If(u) + kg(o)[[
ess sup
6
[[
2[--
f
+ g[l >__ [[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<
Now,
f
that
f(w) + g(0)
I
f +
and
.
I1
are constant
g
f(o %g(l
g(l)li when
+
hf
lira
0
+ xgll-
Loo(,X)
COROLLARY 3.2:
and
Loo(,X)
f
llfll
()
<
2..
Therefore,
llf( o) +
lira
f(mo
f # 0
is a smooth point of
be a finite measure space,
{llf()ll
> ess sup
01
A0
hence
f
such
x()ll-
then the norm function on
e R
X
the norm function on
(ii)
and hence
X
0}
A
is
a Banach space
0
Frchet
is
for
such that
and
Frchet
is
X
Loo(,X)
A
if and only if there exists an atom
f
differentiable at
A
0
This completes the proof of the Theorem.
(R,Z,)
Let
with
li
e AO}
so there exists an
0
a.e. on
ilf!l
and the latter limit eodss since
a mooth point of
-
A
a.e. on
+
llf( O)
gll
(llt() + xm()ll
p
differentiable at
x
0
where
x
0
is
A
0
This follows immediately from the proof of Theorem 3.1.
the essential value of
4.
REPRESENTABLE OPERATORS ON
X
If
Radon-Nikod{m
on
LI(,
is a Banach space and
said to have the
f
(R,E,)
m: Z
X-valued measure
countably additive
absolutely continuous with respect to
/
re(E)
g(0)d()
is. a finite measure space, then
X
X
is
if and only if for every
property with respect to
which is of bounded variation and
g e
there exists a
LI(,X)
such that
E e
for
E
A bounded linear operator
if there exists a
Loo(,X)
g
T(f)
for all
Let
L I(B,o
.
f
LI()
X
is said to be representable if and only
such that
/
f()g(00)d(m)
LI(,E4$
B(LI(,x),X)
into
T
X
denote the Banach space of all bounded linear operators from
For each
o(g)(f)
g e
Loo(,X)
f()g()d()
f
It follows from the results in Diestel and
the
Radon-NikodyCm
Lo(,X)
onto
property with respect to
B(LI(,O,X)
B(L I(,R0
X)
8
B(L I(,,O,X)
Uhl [i, p. 63], that if
then
Frchet
by
el(,aO
o
X
has
is a linear isometry of
Using this fact and Theorem 3.1
characterization of the points of Gateaux and
function on
o(g)
define
we get the following
dlfferentlabillty of the norm
ESSENTIAL SUPREMUM NORM DIFFERENTIABILITY
THEOREM 4.1:
space such that
T e
B(LI(V,0,X
Let
X
X
be a real Banach space and
has the
with
(Frchet)
dlfferentiable at
that
v(A0) < v ()
1
0
(i)
T
a finite measure
Let
property with respect to
The norm function on
B(LI(,O,X)
if and only if there exists an atom
is Gateaux
A
0
for
such
and
1
IITII -0) IIT(XA0) I > (’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