----- Supremum norm differentiability

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A N N A L ES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)

S. H. Cox, Jr. and Sam B. IST

a d l e r

, Jr. (Louisiana) Supremum norm differentiability

1. Introduction. Bestrepo [ 2 ], Smulian [3]-[5], and others have related certain types of differentiability of a norm on a Banach space to some geometric properties (such as separability, uniform convexity, etc.) of the Banach space or its dual. In [ 1 ], p. 168-170, Banach proved that, for the supremum norm )| ||TO on G{X, Bl), where X is a compact metric space and R x denotes the reals,

H m H Z + ^ L - H / l l ,

exists for f , geC{X,RJ)

iff / is a peaking function. In this paper we consider Banach spaces of the form C{X, B) = { /: X B\ f is continuous}, where X is in most cases a compact Hausdorff space and В is a Banach space, with the supremum norm || Ho,,. We give necessary and sufficient conditions that || be Fréchet differentiable at some f e C( X, B). When В itself is endowed with a Fréchet differentiable norm (such as when В is the reals) these conditions can be described completely in terms of topological properties of X. We now give some basic definitions and, in section 2, we state spe

cifically the results described above and indicate some proofs.

Let J b e a set and let (В , || ||B) be a Banach space. A function f : X В is said to be a 'peaking function iff there is a point p e X such that

\\f{p)\\B > \\f{x)\\B for all x eX such that x Ф p; the point p is called the peak point for f and f is said to peak at p.

Let E and F be Banach spaces, let U be an open subset of E, and let / : U -» F. If x0 e Z7, then / is said to be Fréchet differentiable, or simply differentiable, at x() iff there is a continuous linear map l : E -» F such that, for any heE for which (a?0-f&)eZ7, we have:

Wcoix Д)11 f{x0+ h ) —f{x0) = l { h ) X a ) { x 0,h), where lim - — ^ --- = 0 .

Pll-o ||л||

The mapping l is called the differential of f at x0.

2. Throughout the remainder of this paper, unless otherwise stated,

X will denote a compact Hausdorff space, (В , || ||Б) a Banach space,

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and A the Banach space of all continuous functions / : X -> В with the supremum norm || ||TO.

A proof of Lemma 1 can be obtained by modifying part of the proof of a lemma in [ 1 ], p. 168-170.

L emma 1 . If || is differentiable at fe A, then f is a peaking function.

The next lemma characterizes peaking functions in terms of a sta

bility condition in A.

L emma 2. A function fe A peaks at p e X iff, for each neighborhood V

■of p, there exists s > 0 such that g attains its maximum {in norm) in V whenever geA and \\f— дЦ^ < e.

L emma 3. Let fe A such that || differentiable at f, let l denote the differential of || at f, and let p аX be the peak point for f (the point p exists by Lemma 1). I f he A and h vanishes on a neighborhood of p, then l{h) = 0 .

P roof. Let he A such that h is the zero element of В on a neighbor

hood F of p. By Lemma 2 there exists e > 0 such that if 0 < t < e, then f + t - h attains its maximum in norm in V. Since h vanishes on V, this

**implies that ||/+$*й||оо< ll/lloo- However, since**

\ \ f + t - b \ L > \ \ № + t-b m \B = И Ш в = ll/lloo, **it follows that [|/+£A||oo = li/IL Therefore,**

A v **l l / + ^ I L 4 l / L - ï ( * - * ) 0 = lim---!---**

**2-*0+ ll^'^'Hoo**

v - Щ -h) lim ---

<-> 0 + llt'^Hoo = - ЧЬ) , i.e., l{h) — 0 .

T heorem . I f f e A , then || is differentiable at f iff 1 . / peaks at a point j)eZ ;

2 . {p} is an open subset of X;

3 . || ||j 5 is differentiable at f{p).

Moreover, if || is differntiable at /, then ||/||^ = \\f

{ р ) \ \ в ’

epi where p is the peak point for f, ||/||^ and \\f(p)\\'B we the differentials of || ||TO and

|| |[g at f and f(p) respectively, and ep : A В is given by eP (g) = g{p) for each geA.

P roof. Let f e A such that || is differentiable at /. By Lemma 1 , / peaks at a point p e X. To see that {p} is an open'subset of X suppose that {p} is not open and let {pa)af,D be a net in X — {p} such that {pa}aeD -converges to p and f { p a) Ф 0 for each aeD. For each aeD let ha :X ->

[ 0 , 1 ] be continuous such that ha{pa) = 1 and ha vanishes on a neighbor

hood Wa of p and define ga : X -> В by 2h{x)\\f(p)—f( p a)\\B

fa(P)

II f(Pa)\\B

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for all xeX. Using Lemma 3 it can be shown that ]im 11 / Ht» _

IlsU!

However, a simple computation shows that

ll/+ffalloo~ 11/

^H^{o p}

Ы1оо ^> ^1/2

for all a el). This contradiction proves that {p} is an open subset of X.

Next we show that || \\B is differentiable at f{p). Let с : В -> A be the linear isometry (into) given by

(c{b))(x) =

X — p , X Ф p

for each ЪеВ. Let ||/|P and |!c(/(p))jp denote the differentials of || |p at / and c(f(p)j respectively. Using parts 1 and 2 of this theorem it is easy to see that there exists ô > 0 such that if p |p < <5, then

|H /p )) + üj!oo = \\c(f(p))(p) + h(p)\\B = \\f{p) + h(p)\\B and

ll/4-АЦоо =

\ \ f ( P ) + h { p ) \ \ B .

It follows that

lim 4 f w j + A | U (/(î>))i|oo — ll/lL(A)

which proves that || IP is differentiable at c{f(p)) and, in fact, ||c(/(p ))||^0

— Il/L- Since c is a continuous linear mapping of В into A, c is differen

tiable at f(p). Hence, since the diagram В X A -

il Hi?

----> Bl

loo

commutes, the chain rule gives that || \\B is differentiable at f{p). In fact the differential \\f(p)\\B of || ||B at f(p) is given by the formula \\f(p)\\B

— ||/L ° c . This completes the proof of the first half of the theorem.

To prove the second half of the theorem assume 1 , 2 , and 3 hold for

/

and let

e > 0 .

There exists

<5 > 0

such that if he A and

P I P <

Ô, then IL m iL = \\f(p) + h(p)\\B, and

М { р ) + ч р ) \ \ в - м т \ в - \ \ п р ) \ Ш р ) )

\\4 v )\\ b

R oczn ik i PTM — P ra ce M a tem a ty czn e XV 9

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where \\f(p)\\в is the differential of || \\B at f ( p ). It follows that if

< <5, then

ll/+ A IL H I/ll» H I/( p ) lle ( * W ) „

--- !--- — < £ . P I L

This proves that || [(^ is differentiable at f and, moreover, verifies the formula ||/||^ = \\f (p)\\'Bo ep ■ This completes the proof of the theorem.

In the following three corollaries the Banach space (B, || j|5) is specifically taken to be the real numbers R 1. However, the corollaries remain valid if the real numbers is replaced by a Banach space with a differentiable norm (i.e., with a norm differentiable on the space without zero).

C orollary 1 . The compact Hausdorff space X is perfect iff || is nowhere differentiable on G(X, R 1).

C orollary 2. I f X is a compact Hausdorff space such that each point is a Gô, then || is differentiable at every peaking function in C ( X , R 1) iff X has only a finite number of points.

C orollary 3. I f X is a compact Hausdorff space, then || is differen

tiable on a dense open subspace of С ( Х , Н г) iff {xeX\ x is a limit point of X } is nowhere dense in X.

Proof. Let D — { f eC( X, R^l f peaks at an isolated point of X }.

The set D is open in X and, by the Theorem, D is precisely the points of differentiability of || . It is not -difficult to see that D is dense in C(X, R 1) iff {xeX\ x is a limit point of X} is nowhere dense in X.

It is easy to give an example of a compact Hausdorff space X with an infinite number of points such that || is differentiable at every peaking function in C(X, R 1). In fact, this differentiability condition is vacuously satisfied when X is a space in which no point is a Gô (for example, the uncountable product of unit intervals). However, there are examples of compact Hausdorff spaces X with an infinite number of points such that || differentiable at every peaking function in G ( X , R 1}

and the peaking functions form a dense open subspace of C(X, R l).

Using the Theorem and Corollary 3 it is easy to see that this is the case if X is the one point compactification of an uncountable discrete set.

We now give a result concerning the differentiability of the supremum norm on spaces of bounded real-valued functions defined on a locally compact Hausdorff space.

C orollary 4. Let X be a locally compact Hausdorff space and let

B ( X , R 1) = { / : X R x\f is continuous and bounded} with the supremum

norm || ||jg. If f e B { X , R1), then |[ (|B is differentiable at f iff

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1 . f peaks ar a point peX-, 2 . {p} is an open subset of X;

3. \f(p)\ is bounded away from {\f(cc)\\coe(X-~{p})\.

Proof. Let /ЦХ) be the Stone-Cech. compactification of X and define <p: B { X , B 1) -> C(fi(X), Æ1), by extension. Since <p is an isometric isomorphism of B { X , B l) onto C((3(X), B1), it follows that [| \\B is dif

ferentiable at / iff the snpremnm norm on C((3(X), B 1) is differentiable at cp(f). The proof may now be completed by applying the Theorem several times together with the fact that a point of X is open in X iff it is open as a point of (Ï(X).

Bibliography

[1] S. B a n a c h , Théorie des opérations linéaires, Warszawa 1932.

[2] G. R e s tr e p o , Differentiable norms in Banach spaces, Bull. Amer. Math. Soc.

70 (1964), p. 413-414.

[3] У. S m u lia n , On some geometrical properties of the sphere in a space of the type (B), Comptes Rendus (Doklady) de FAcademie des Sciences de l ’URSS 24 (1939), p. 648-652.

[4] — Sur la dérivalilite de la norme dans Vespace de Banach, ibidem 27 (1940), p.

643-648.

[5] — Sur la structure de la sphère unitaire dans Vespace de Banach, Mat. Sbornik 9 (1941), p. 545-560.

L O U ISIA N A STATE U N IV E R S IT Y BATON ROUGE, L O U ISIA N A