SUPREMUM NORM DIFFERENTIABILITY

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 Z(E) of all bounded sequences in E and to the space B(Z I, E) of all bounded linear operators from Z into E

This theorem, however, is no longer true if T is a locally compact, non- 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 is the set of positive integers equipped with the discrete topology, then l C (N) the space of all bounded continuous functions on IN n for n > then If we let x {Xn}n_>l l where x and Xn n x peaks at n but because of the behaviour of x at infinity and the existence of Banach limits, it is possible to find two distinct support functionals to the ball in Z at x so that x is not a smooth point.
In this note, we characterize the points of Gateaux and Frchet differenti- ability of the norm function in C(T, E) the space of all bounded continuous E-valued functions on the locally compact Hausdorff space T where E is a real Banach space.
Two applications of these results are given.The first is to the space Z(E) of all bounded sequences in E and the second to the space B(Z I, E) of all bounded linear operators from Z1 into E 2.
In the following, E denotes a real Banach space and E* denotes the dual of E llxll I} is the unit sphere of E A Banach space E is said to be smooth at x E {0} if and only if there X exists a unique hyperplane of support to B E at T that is, there exists only one continuous linear functional q E* with II@II such that p(x) llxli x Such a linear functional ap E* is called the SUpport functional to B E at ]T X and p I({I}) is called the hyperplane of support to B E at x-IT A Banach space E is said to be a smooth Banach space if it is smooth at every x S E

SMOOTH POINTS IN C(T, E)
If T is a topological space and E is a real Banach space, then C(T, E) denotes the Banach space of all bounded continuous E-valued functions on T with the supremum norm; that is, C(T, E) {f: T E f is bounded and continuous} and Jjfj sup {jf(t)J t 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 f 0 if and only if f is a peaking function, that is, the exists a point t 0 T such that jjf if(to) > Jf(t)J for all toT, 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) when T is compact Hausdorff.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) when T is locally compact Hausdorff.The techniques used by Cox and Nadler will not work in this case, since, in general, the range of f c C(T, E) is no longer relatively compact and hence no extension to an C(6T, E) is possible.However, a slight modification of the argument given in K6the [7], (page 352), for the corresponding sult in Z does work.
Frchet differentiability follow as a corollary such that llf(to)ll llf(tl)ll For t T and 0 E* let aO,t C(T, E)* denote the evaluation functional given by O,t(g) o(g(t)) for g C(T, E) then I160,tI for all t T 0 SE, Using the Hahn-Banach theorem, choose 00, 01 E* with 10011 IlOll such that O0(f(to) l(f(tl) Then 600,t 0 and Ol,t are distinct support functionals to the ball in C(T, E) at f which contradicts the fact that f is a smooth point in C(T, E) (II) We show next that given any compact set K T either sup llf(t)ll < or there exists a t O T K such that llf(to)ll tT~K To the contrary, suppose there exists a compact set K c_T such that sup IIf(t)II and [If(t)ll < for all t T K Let Fo(t llf(t)ll tT~K for all t T Then F 0 is a bounded continuous function on T and thus has an extension, F to T the Stone-ech compactification of T Since sup llf(t)ll and thus A is a Ga set in BT By ech [2], singletons in BT~T are not Ga sets, so A contains at least two distinct points p and q Let {pu} and {qu} be disjoint nets contained in T such that pu p and q q in T For each let E* with Ilqbll and @u(f(pu)) F(pu) Also, for each u choose c E* with llull and v (f(q) F(q) Let and for each and ,q ,P Then @u and C(T, E)* and II@ull llull for each u and Since the ball in C(T, E)* is w*-compact, there exist @, C(T, E)* with II@ll <-and IIII <_ such that @ is a w*-accumulation point of the net {9 and is a w*-accumulation point of the net { By construction, @(f) (f) and, thus, @ and are support functionals to the ball in C(T, E) at f Since f is assumed to be a smooth point in C(T, E) it must be that @ We will show that this is impossible Let P {pu}U{p} and Q {qu} U {q} Then P and Q are disjoint closed subsets of T which is a compact Hausdorff space and therefore normal.Let h I, h 2 C(BT) with 0 < h I, h 2 _< I, h + h2 and hl (P) h2(Q) 0 Use h for the restriction of h to T as well.
Clearly, if g C(T, E) then hlg ker @ and h2g ker Since @ , and g C(T, E) can be written as g hlg + h2g we have @ 0 But, this contradicts IIII llII Therefore, we must have that either sup llf(t)ll < or there exists t O T K with Ilf(t 0)II for any tT~K compact set K_c T (III) Finally we show that (i__), (i__i), (iii) hold Taking K I in (II), since llfll we see that there exists a t O T with llf(to)ll and from (I), llf(to)ll > llf(t)ll for all t t O Again by (II), if K T is a compact set with t O K then sup llf(t)ll < If there exist distinct functionals tT-K 01' @2 E* with llqll llp211 such that @l(f(to) qb2(f(to) then C(T, E)* are distinct support functionals to this implies that 6@l,tO, @2,to the ball in C(T, E) at f which contradicts the fact that f is a smooth point.
Therefore f(to) is a smooth point of E B.. Conversely, suppose that f C(T, E) llfll and (i), (ii), and (iii) hold; then there exists a unique t o T such that llf(t 0)II there exists a compact set K:_ T with t O K such that sup llf(t)II < and E is smooth tT~K at f(to) Let g C(T, E) g 0 and let 6 > 0 be such that llf(t)ll < IIf(t O) II-6 for all tT~K If 0 < II < 6 then for t T K we have Thus, f(t) + g(t) < f(to) for all t T K whenever 0 < < On the other hand, Therefore, for 0 < < a 21lgl sup llf(t)+ ),g(t)II sup llf(t)+ g(t)ll tT tK Since K is compact, by Kondagunta's result [6] sup IIf(t) + g(

B(Z I, E)
Let E be a Banach space, let Z be the Banach space of all absolutely summable real valued sequences with llall lanl for n=l a {an}n_l Z and let B(Z I, E) be the space of all bounded linear operators from Z into E For n -> let n be the n th basis vector in Z that is n {6n k k>l THEOREM 4.2.
Let E be a Banach space, then the norm function ll-II B(Z I, E)/ IR + is Gateaux (Frchet) differentiable at T B(Z I, E)   if and only if n O (i) there exists an n O such that IIT( )If > llT(n)ll for n n O (ii) sup llT(n)ll < IITII nn 0 (iii) the norm function II-II E /IR + is Gateaux (Frchet)   n O differentiable at T(6 PROOF.The mapping " B(Z I, E) Z(E) given by o(T) {T(n)}n>_l for T B(ZI, E) is a linear isometry of 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 differentiability of the norm in K(E, F) the space of compact linear operators from E into F where E and F are Banach spaces.He has also given a characterization of the points of Frchet differentiability of the norm in B(E, F) the space of bounded linear operators from E into F (no proofs are given in this paper).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, C(T, E) (or C(T, R)) is smooth at f if and only if f peaks at some t O T (See e.g.Holmes [4], p. 232, #4.10).
A result which is obviously false, as the example in the intr(duction demonstrates.