Internat. J. Math. & Math. Sci.
Vol. 6 No. 4 (1983) 705-713
7O5
SUPREMUM NORM DIFFERENTIABILITY
I.E. LEONARD and K.F. TAYLOR
Department of Mathematics, University of Saskatchewan,
Saskatoon, Saskatchewan, Canada S7N OWO
(Received June I, 1982)
ABSTRACT.
The points of Gateaux and Frchet differentiability of the norm in
C(T, E) are obtained, where T is a locally compact Hausdorff space and E is a
real Banach space.
Applications of these results are given to the space
all bounded sequences in
E
operators from Z
E
into
and to the space
Z(E)
of
B(Z I, E) of all bounded linear
KEY WORDS AND PHRASES. Banach space, continuo functions, vector-valued functions,
supremum norm, smooth points.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
46B20,
Primary:
46E40.
Secondary: 46A32, 46A45.
1.
INTRODUCTION.
In [I], Banach proved that if T is a compact metric space and C(T)
Banach space of all continuous real valued functions on
T
with the supremum norm,
then
im
Ilf + ,g]]-
I./0
exists for all
If(to)
>
If(t)
g
C(T)
for all
if and only if there exists a
t
T
t
tO
tO
is the
T such that
706
I. E. LEONARD AND K. F. TAYLOR
T is a locally compact, non-
This theorem, however, is no longer true if
compact, Hausdorff space; as can easily be seen by considering the Banach space
of all bounded real valued sequences with the supremum norm.
In fact, if IN
topology, then
If we let
x
x peaks at
n
is the set of positive integers equipped with the discrete
of all bounded continuous functions on IN
n
then
for n >
and
x
n
but because of the behaviour of x at infinity and the
l C (N) the space
{Xn}n_>l l where
Xn
existence of Banach limits, it is possible to find two distinct support functionals
to the ball in
Z
at x
is not a smooth point.
x
so that
In this note, we characterize the points of Gateaux and Frchet differenti-
C(T, E)
ability of the norm function in
the space of all bounded continuous
T
E-valued functions on the locally compact Hausdorff space
E
where
is a real
Banach space.
The first is to the space
Two applications of these results are given.
E
of all bounded sequences in
bounded linear operators from
E
into
DEFIN TIO.NS AND IOTATI ON.
2.
In the following,
E
S
The unit ball of
{x
E
E
llxll
E
E
I}
A Banach space E
denotes a real Banach space and
is
BE
is the unit sphere of
I({I})
B E at
with
E
x
if and only if there
{0}
X
that is, there exists only
T
II@II
such that
ap
E*
E
p(x)
llxli
is called the
x-IT
is said to be a smooth Banach space if it is smooth at every
The norm function
x
E*
and its boundary
x
SUpport functional to B E at ]T
X
is called the hyperplane of support to B
A Banach space
E at
Such a linear functional
p
q
I}
denotes the dual of
E*
E
is said to be smooth at
one continuous linear functional
and
xll -<
E
{x
exists a unique hyperplane of support to
E
B(Z I, E) of all
and the second to the space
Z1
Z(E)
ll-li
E-FR
+
llx
S
E
is said to be Gateaux differentiable at
{0} if and only if there exists a functional
lira
x
+
>hti- llxjl
>
#
(h)
E*
ith
o
(*)
SUPREMUM NOI DIFFERENTIABILITY
for every
at x
E
x
E
at
E
h
]m
fix. +._hil
+
E +;R
.J’"
3.
__(.h)
exists for all
s
E
such that
(**)
BE
smooh aZ
E
h
E*
0
(*) exists uniformly for h
E
x
f and only
{0}
f and only f he norm function
is Gateaux diffentiable at
SMOOTH POINTS IN
If
jixll
well known, Mazur [8], ha
xhJJ- llxll
jlx +
Frchet differentiable
is said to be
if and only if there exists a functional
that is, the limit in
s
I+
E
The norm function
im
i
is called the Gateaux derivative of the norm
The functional
{0}
707
x
C(T, E)
T is a topological space and E
is a real Banach space, then
denotes the Banach space of all bounded continuous E-valued functions on
C(T, E)
T
with
the supremum norm; that is,
C(T, E)
and
Jjfj
{f: T
sup {jf(t)J
E
f
t
is bounded and continuous}
T}
for
f
s
C(T, E)
As mentioned earlier, Banach [I] proved that if T is a compact metric space,
then
C(T)
C(T, )
is smooth at
that is, the exists a point
toT,
t
0
f
if and only if
0
T such that
jjf
f
is a peaking function,
if(to)
>
Jf(t)J for
all
tt 0
Kondagunta [6], and Cox and Nadler [3], have characterized the points of
Gateaux and Fr6chet differentiability of the nora in C(T, E)
Hausdorff.
when
T
is compact
Cox and Nadler, in the same paper, give a characterization of the points
of Fchet differentiability of the norm in
C(T, ) when T
is locally compact
Hausdorff.
In this section, we generalize these sults to the space C(T, E)
is locally compact Hausdorff.
when
T
The techniques used by Cox and Nadler will not work
in this case, since, in general, the range of
compact and hence no extension to an
f c C(T, E)
C(6T, E)
is no longer relatively
is possible.
However, a slight
modification of the argument given in K6the [7], (page 352), for the corresponding
sult in
Z
does work.
708
I.E. LEONARD AND K.F. TAYLOR
As usual, the results on smoothness are the more difficult, and the results on
Frchet differentiability follow as a corollary
THEOREM 3I
space
Let T be a locally compact Hausdorff space and E a Banach
A point f
(!) there
C(T, E), f
exists a
O,
is a smooth point of
T such that
tO
C(T, E)
llf(to)ll > llf(t)ll
if and only if
for all
T,
t
tt 0
(ii) there exists a compact neighborhood K of t O such that
sup
tT~K
(ii___i)
f(to)
llf(t)ll
llfll
<
is a smooth point of
E
PROOF.
A
f
II.II
Assume first that the norm
C(T, E)
C(T, E)
[R
where we may (and do) assume that
+
is Gateaux differentiable at
IIfll
T R+ achieves its maximum
(I) we show first that if the mapping IIf(.)ll
at t O T
such that
then
tO
is unique.
llf(to)ll llf(tl)ll
Suppose there exist t t
T, t O
O
For t T and 0 E* let aO,t
denote the evaluation functional given by
I160,tI
then
choose
Then
at
for all
E* with
00, 01
and
600,t 0
f
T
t
O,t(g)
SE,
0
10011 IlOll
o(g(t))
llf(t)ll
Ol,t
or there exists a
<
O0(f(to)
l(f(tl)
f
T
tO
K
sup
for all
IIf(t)II
t
an extension,
sup
teT-K
llf(t)ll
and
T
Then
F
[If(t)ll
<
t
for all
either
such that
T
C(T, E)
C(T, E)
is a smooth point in
llf(to)ll
To the contrary, suppose there exists a compact set K c_T
tT~K
C(T, E)
g
are distinct support functionals to the ball in
(II) We show next that given any compact set K T
sup
C(T, E)*
Using the Hahn-Banach theorem,
such that
which contradicts the fact that
tT~K
for
tl,
K
such that
Fo(t
Let
llf(t)ll
F0 is a bounded continuous function on T and thus has
to T
we have
the
A
A
Stone-ech
compactification of
F -l(1) CI(T
/ {t
n=l
T
T)
K
Also
0
F(t)
T
>
l_}
n
Since
SUPREMUM NORM DIFFERENTIABILITY
A is a
and thus
Ga
BT
set in
ech
By
[2], singletons in BT~T are not
sets, so A contains at least two distinct points
Let
and
{pu}
q
@u(f(pu))
v (f(q)
q
Then
and
and
@u
For each
II@ll <-
{9
and
@(f)
let
F(pu)
Also, for each u
F(q)
Let
is a
(f)
and
Ilqbll
E*
with
for each
<_
such that
is a
@
Since
be that
@
We will show that this is impossible
Q
U {q}
Then
{qu}
and
u
C(T, E)*
@,
C(T, E)
is assumed to be a smooth point in
P
and
By construction,
{
C(T, E) at f
f
and
are support functionals to the ball in
and
@
llull
w*-accumulation point of the net
w*-accumulation point of the net
and, thus,
Let P
it must
{pu}U{p}
Q are disjoint closed subsets of T
and
p
and
for each
,q
II@ull llull
pu
T such that
with
c
Ga
q
and
C(T, E)* is w*-compact, there exist
IIII
and
E*
choose
,P
C(T, E)* and
Since the ball in
with
p
be disjoint nets contained in
{qu}
T
in
709
and
which is a
compact Hausdorff space and therefore normal.
<
h I, h 2 _< I, h +
of h
to T as well.
0
h2
Clearly, if
and
IIII llII
contradicts
llf(t)ll
(III)
Finally we show that
llfll
(I),
hlg
we see that there exists a
t
K
then
sup
tT-K
llf(t)ll
llqll
llp211
tO
E* with
this implies that
the ball in
Therefore
h2g
h2g
ker
we have
Since
0
@
K
with
Ilf(t 0)II
@
,
But, this
for any
6@l,tO,
C(T, E) at f
f(to)
@2,to
I
(i__), (i__i), (iii) hold Taking K
for all
01’ @2
T
tO
llf(to)ll > llf(t)ll
set with
+
and
@
Therefore, we must have that either
or there exists
<
ker
K_c T
compact set
since
hlg
then
C(T, E) can be written as g
g
sup
tT~K
hl (P)
and
C(T, E)
g
Let h I, h 2 C(BT) with
h2(Q) 0 Use h for the restriction
tO
tO
T with
llf(to)ll
Again by (II), if
<
such that
in (II),
and from
K T is a compact
If there exist distinct functionals
@l(f(to)
qb2(f(to)
then
C(T, E)* are distinct support functionals to
which contradicts the fact that
is a smooth point of
E
f
is a smooth point.
I. E. LEONARD AND K. F. TAYLOR
710
B..
Conversely, suppose that
hold; then there exists a unique
compact set
at
K:_ T
and (i),
(ii),
T such that llf(t 0)II
to
K
tO
with
llfll
C(T, E)
f
llf(t)II
sup
such that
there exists a
<
and
be such that
llf(t)ll
tT~K
and (iii)
E is
smooth
f(to)
C(T, E)
Let g
for all
tT~K
f(t)
Thus,
g
If 0
g(t)
+
0
<
and let
II
<
f(to)
<
6 > 0
6
T
t
for all
T
t
then for
K
<
IIf(t O) II-
6
we have
K whenever 0
<
<
On the other hand,
Therefore, for
<
<
a
21lgl
llf(t)+ ),g(t)II
sup
tT
Since
0
sup
tK
llf(t)+ g(t)ll
K is compact, by Kondagunta’s result [6]
IIf(t)
sup
tK
lim
g(t)II- sup llf(t)II
+
tK
IIf
lim
+0
t xgil-
Ilfll
/0
exists for all
C(T, E)
g
Hence C(T, E)
is smooth at
f
q.e.d.
An analogous result holds for the Frchet differentiability of the norm in
C(T, E)
COROLLARY 3.2.
space.
f
Let
T be a locally compact Hausdorff space and E
The norm function
C(T, E)
f
0
II.I!
of
{t
O
is
a Banach
Frchet differentiable at
if and only if
(i) there exists a unique
(ii)
C(T, E)/ IR
+
T such that
to
T
is an open subset of
that is,
IIf(to)ll >
t
O
sup
tt 0
llf(t)ll
is an isolated point
T
(iii) the norm function
(Note"
(ii) follows from (i)
II-I
E
/[R
+
is Frchet differentiable at
f(to)
SUPREMUM NORM DIFFERENTIABILITY
711
PROOF
A
f
Suppose that
s
tO
C(T E)
llfll
C(T, E),
exists a
I I
s
sup
tT-K
/
is Frchet differentiable at
C(T, E) is smooth at f
then the ball in
llf(t)II
T
t
for all
so there
K of t O such that
T and a compact neighborhood
(I) llf(t 0)II > ]]f(t)ll
(2)
IR
/
t
to
<
(3) E is smooth at f(t O)
Now
lim
Ill
+ II Ilfll
lim
>, /0
supllf(t) + xg(t)ll
supllf(t)ll
tK
tK
,/0
exists uniformly for
g
BC(T,
result of Cox and Nadler [3] shows that
Fr6chet differentiable at
f(to)
compact, the uniqueness of t O
K is compact, an appeal to the
and since
E)
{t
II.II
is open and
O
Also, since
{t
sup
shows that
tt 0
is open and
O
llf(t)ll
+
E /IR
K
{t
is
O
is
<
Conversely, suppose that (i), (ii), and (iii) hold. Using the previous theorem,
+
and taking
is Gateaux differentiable at f
we see that ll.II
C(T, E)/JR
B.
K
{t
0}
lim
IIf + xgll
llfll
lim
llf(t)II
/o
/0
exists uniformly for
able at
llf(to) + xg(t)II
g
BC(T,
I I
E) since
E/ [R
+
is Frchet differenti-
f(t O)
q .e .d.
4.
APPLICATIONS.
A.
l(E)
Z (E)
If
{x
E
is a Banach space, then
{Xn}n>iI
with the supremum norm,
xll
xn
c
E
n _>
for
and
sup
n>_l
sup
n>_l
xn
I
llxnll
<
I. E. LEONARD AND K. F. TAYLOR
712
THEOREM 4.1.
+
/IR
Let
II.II (E)
E be a Banach space, then the norm function
is Gateaux
(Frchet) differentiable at x
{Xn}n_>l,
x
0
if and only if
(i) there exists an n O such that
nn 0 llXnl
(ii)
sup
PROOF.
xn
Let IN
for n
no
II-II
E/
IR+
is Gateaux
(Fr6chet)
O
denote the set of positive integers equipped with the discrete
Z(E)
topology, then
llXnl
>
llxll
<
(iii) the norm function
differentiable at
llXnoll
C([N, E)
the space of bounded continuous
functions on the locally compact Hausdorff space
E-valued
N
q.e.d.
B.
B(ZI, E)
Let E
be a Banach space, let
absolutely summable real valued sequences with
a
from Z
into
that is
n
llall
lanl
n
E
{6nk
THEOREM 4.2.
ll-II
be the Banach space of all
for
n=l
and let B(Z I, E) be the space of all bounded linear operators
For n ->
let
be the n th basis vector in Z
Z
{an}n_l
Z
B(Z I, E)/ IR
k>l
Let
+
E
be a Banach space, then the norm function
is Gateaux
(Frchet) differentiable at T
B(Z I, E)
if and only if
n
(i) there exists an n O such that IIT( O )If
(ii)
sup
nn 0
PROOF.
for
T
B(ZI,
T(6
nO
The mapping
E)
llT(n)ll
for
n
nO
llT(n)ll < IITII
(iii) the norm function
differentiable at
>
"
II-II
E /IR +
B(Z I, E)
is Gateaux
Z(E)
is a linear isometry of
given by
(Frchet)
o(T)
{T(n)}n>_l
B(Z I, E) onto Z(E)
REMARKS.
I.
In connection with the second example, it should be mentioned that Kheinrikh
[5] has given a complete characterization of the points of Gateaux and Fr6chet
SUPREMUM NORM DIFFERENTIABILITY
differentiability of the norm in
from
E
F
into
where
K(E, F)
713
the space of compact linear operators
E and F are Banach spaces.
He has also given a
characterization of the points of Frchet differentiability of the norm in
the space of bounded linear operators from
this paper).
E into
F
B(E, F)
(no proofs are given in
However, the more difficult question of smoothness in
B(E, F)
is
still unanswered.
2.
Regarding Theorem 3.1. )erhaps this will clear up the popular misconception
that, for T
locally compact Hausdorff,
if and only if
f peaks at some
T
tO
C(T, E)
(or C(T, R))
is smooth at
f
(See e.g. Holmes [4], p. 232, #4.10).
A result which is obviously false, as the example in the
intr(duction demonstrates.
RE FE REN CE S
Thorie des Operations Linaies, Monografie Mat.
I__, Warsaw, 1932.
I.
BANACH, S.
2.
CECIl, E.
3.
COX, S.H. Jr. and S.B. NADLER, Jr. "Supremum norm differentiability"
Ann. SOCo Math. Polonae 15, 127-131, 1971.
4.
HOLMES, R.B.
Geometric Functional Analysis and its Applications, Graduate
Texts in Mathematics 24, Springer-Verlag, New York, 1975.
5.
KHEINRIKH, S.
6.
KONDAGUNTA, W.
"On bicompact spaces" Ann. of Math. 38, 823-844, 1937.
"The differentiability of the norm in spaces of operators"
Funkcional Anal.
Prilozen 9 (4), 93-94, 1975. English translation"
Functional Anal. Appl. 9__ (4) 360-362, 1975.
"Some geometric properties of the unit cell in the space
Bull. Acad. Polon. Sci., Ser. Math. Astron. Phys. 19 (II),
1007-1012, 1971.
C(X, B)
7.
K(THE, G.
8.
MAZUR, S.
Topologische linear Rume I, Die grundlehren der math. Wissenschaften 107, Springer-Verlag, Berlin, 1966. English translation"
Die grundlehren der math. Wissenschaften 159, Springer-Verlag, New York,
1969.
"Ober konvexe Mengen in
4, 70-84, 1933.
inearen normierten Rumen"
Studia Math.