ESSENTIAL SUPREMUM NORM DIFFERENTIABILITY

The points of Gateaux and Frechet differentiability in L∞(μ,X) are obtained, where (Ω,∑,μ) is a finite measure space and X is a real Banach space. An application of these results is given to the space B(L1(μ,ℝ),X) of all bounded linear operators from L1(μ,ℝ) into X.


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 ess sup {If(t) l: t e [0,I]} (as usual, identifying functions that agree a.e. on [0,i]) In [4], Mazur proved that given any f e L(m,RR) f # 0 there exists a g L(m,0 such that lira jjf + gjj ijfjj does not exist.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 L (,X) the space of ;J-measurable, essentially bounded functions, whose values lie in a Banach space X-provided that the underlying measure space (,Z,) is non-atomic.We then obtain a complete description of the smooth points of L=(,X) in the general case.We show, in fact, that f is a smooth point of L(,X) if and only if f achieves its norm on a unique atom for and its (-a.e.constant) value on this atom is a smooth point of X operators rom LI(,0 into a Banach space X when X has the Randon-Nikodym property with respect to

PRELIMINARIES
Throughout this note, X denotes a real Banach space with dual X* A point x X {0} is a smooth point of X if there exists a unique e X with IIIi 1 such that (x) llxll The norm function on X is Gateaux differentiable at non-zero x e X if there exists a X* such that liml[lx + ,hll ilxll O(h) 0 (*) for all h e X 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 a smooth point of X (ii) lira llx + %hll-llxll exists for all h e X (iii) the norm function on X is Gateaux differentiable at x The norm function on X is Frchet differentiable, at a non-zero x e X if there exists a e X* such that Of course, Frchet differentiability at a point implies Gateaux differentiability at the point.
Let (Q,Z,) denote a finite measure space.A mapping f: X is called measurable (or strongly measurable) if -i (i) f (V) e Z for each open set V = X and (ii) f is essentially separably valued; that is, there exists a set N e Z with (N) 0 and a co,table set H = X such that f( N) = H The Lebesgue-Bochner function space L=(,X) is the real vector space of all -measurable, essentially bounded, X-valued fctions defined on L=(,X) is a real Banach space when equipped with the norm f ess sup {[f()[[: } (as usual, identifying functions which agree v-a.e.) A set A e Z is an atom for the measure V if and only if v(A) > 0 and for any B e Z with B = A either v(B) 0 or v(B) (A) The measure space (9,Z,) is called non-atomic if there are no atoms for V in Z and rely atomic if Q can be expressed as a ion of atoms for We will write O 9d with Q d e Z for the (essentially unique) decomposition of c c into its non-atomic and purely atomic parts.Since 0 is a finite masure, there exists an at st countable pairwise disjoint collection {A i i e I} of atoms for such that d A.
We note that if A is an atom for V and f e L (v,X) for p We will need the following lemmas in the discussion of the smooth points of L,(, X) Xn are Banach spaces, then (x I, x 2, Xn is smooth point of (X 1 X 2 Xn ) if and only if there exists a Jo n such that Jo-(i) llxj > llxjll for j 4 Jo and 0 (ii) x.
is a mooth point of X.
If (d' ld' d and (c, Zc' c are the purely atomic and non-atomic measure spaces, respectively in the decomposition of fl" then L (,X) is isometrically isomorphic x)) to (L=(d,X) L(U c, The proof of the second lemma is routine, while the proof of the first lemma uses the fact that (X 1 X 2 Xn) is isometrically isomorphic to (X 1 @ X 2 E) Xn) 1 see [3] The next two lemmas are straightforward generalizations of results in Kothe [2], we sketch the proof of the first lemma.
LEMMA 2.3: Let X be a Banach space and let E(X) denote the space of bounded sequences in X with the supremum norm.If x {Xn }n>l (X) x # 0 Lhen x is a smooth point of =o(X) f and only if there exists a positive integer n o such that (i) llxn0ll > up {llXnl n @ n O} and (ii) x is a smooth point of X n O PROOF.
Let x (x (X) be a smooth point we may assume that llxll sup llXnl i n n_l n>!If there exists a mubsequence {xnk}k> I__ such that k-lira llxnkll--I we can demonstrate the existence of distinct elements of E(X) which support the unit ball at x by the following modification of the argument given in Kthe [2] for (0 We consider the disjoint sequences {x and {x For each n 2j j>_l n2j-i j_>l j _> i let j and Sj be elements of X such that lljll lljil i with x llx n I I and (x llXnmj_lli J( n2j 2j 'bJ n2j-i Define .jand ?.3 on ,(X) by j (y) j (Yn2 j and ''j (y) j (Yn2j-l) for all y {Yn}n>l e (X) and j i then ., '. e 9 (X) and ]= j[ for.
z Let , and be w*-accumulation points of the sequences {+j }jl and {' j}jl respectively, then by construction we have This contradicts the fact that x is a smooth point of (X) us, we have sho that if x {Xn}n>l is a smooth point of (X) then li Xn < [x[ and therefore there must exist a positive integer n 0 such that n, [Xn0{ x] If there exists another integer m 0 # n 0 with Xmo ][x let 1' e X with 1 i and (0 (Xm0) x Now define P e (X) by #(y) #(Yn0) d V(y) (Ym0) for y {Yn}nl e (X) then + and P are distinct support functionals to the ball in (X) at x Again, a contradiction.We have established that if x is a smooth point of (X) then (1) must hold.A similar argent shows that (ll) must hold as well.
Conversely, If x {Xn}n>l e (X) and (1) and (i) hold, then for any y {Yn}nl e (X) y # 0 we have x + %y Xn0 + %Yn0 for all A e satisfying [[ (x-sup{ n # n0}) Therefore, IXn 0 Yn 0 Xn 0 m x + Zl[-[ im which exists by (ll); them, x Is a smooth point of m(X) This completes the proof of the lena.
#m argent similar to the above gives the following: LEMMA 2.4: Let (,Z,) be a finite measure space which is purely non-atomlc, and let X 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 L(,X) THEOREM 3.1: Let (,Z,U) be a finite measure space, X a Banach space, and f e L(,X) with f @ 0 then f is a smooth point of Lm(,X) if and only if there exists an atom A 0 for such that (i) llfli ess sup {ilf(o)]l e A O} and (ii) x 0 is a mooth point of X where x 0 is the essential value of f on A 0 PROOF.
Suppose f e L(v,X) f 0 is a smooth point of Lo=(v,X) then Lemma 2.4 implies that Z contains at least one atom for V Let fl fc U rid be the decomposition of r into its non-atomic and purely atomic parts.Since, by Lemma 2.2, Loo(v,X) is isometrically isomorphic to (L (Vc' and the fact that f i a smooth point of L(v,X) imply that either I" llfl > es sup {llf()ll 6 rid and x) fl is a mooth point of L (Vc' c 2 o. l[flzdll ess up llf(>ll =zc and fld is a .moothpoint of L(d,X) Now, ease 1 i ru!ed out by Lena 2.4, since (Q ,Z , ib a finite on-atomic c c c measure space.Therefore, l]fl?
Let d U A i where {A.i e I} is a paiise disjoint collection of atoms for since is finite, then either I is finite or cotably infinite.
I i finite, then m(Ud,X) is isometrically isomorphic to (XIX..) with X.
X for l, 2 n while if I is countably infinite, then L (d,X) is isometrically isomorphic to (X) In either case, it is easily seen (from Lemma 2.1 or Lena 2.3) that there exists an atom A 0 for with f > up f(> n A0 =d (ii) x 0 is a smooth point of X where x 0 is the essential value of f on A 0 Conversely, suppose that f 8 L(,X) and there exists an atom A 0 for Z such that (i) and (ii) hold.Let m0 AO with f(mo x 0 then from (i) we have lf[ f(0 Now, f and g are constant a.e. on A 0 so there exists an 01 A 0 such that f(w) + g(0) f(o + %g(la.e. on A 0 and hence I I f + gll llf( O) + g(l)li when 0 li < 2.. Therefore, llf( o) + x()lllira hf + xgll-ilf!llira and the latter limit eodss since f(mo is a smooth point of X hence f is a mooth point of Loo(,X) This completes the proof of the Theorem.
COROLLARY 3.2: Let (R,Z,) be a finite measure space, X a Banach space and f Loo(,X) with f # 0 then the norm function on Loo(,X) is Frchet differentiable at f if and only if there exists an atom A 0 for such that () llfll > ess sup {llf()ll e R A 0} and (ii) the norm function on X is Frchet differentiable at x 0 where x 0 is the essential value of f on A 0 This follows immediately from the proof of Theorem 3.1. 4. REPRESENTABLE OPERATORS ON LI(, If X is a Banach space and (R,E,) is. a finite measure space, then X is said to have the Radon-Nikod{m property with respect to if and only if for every countably additive X-valued measure m: Z X which is of bounded variation and absolutely continuous with respect to there exists a g e LI(,X) such that re(E) / g(0)d() for E e E A bounded linear operator T LI() X is said to be representable if and only if there exists a g Loo(,X) such that T(f) / f()g(00)d(m) for all f .LI(,E4$ Let B(LI(,x),X) denote the Banach space of all bounded linear operators from L I(B,o into X For each g e Loo(,X) define o(g) 8 B(L I(,,O,X) by Let T e B(LI(V,0,X with T $ 0 The norm function on B(LI(,O,X) is Gateaux (Frchet) dlfferentiable at T if and only if there exists an atom A 0 for such that 0 v(A 0) < v () and 1 1 (i) IITII -0)IIT(XA0) I I > ('flA0 IIT(XA 0) I I and (ii) T(XA O) is a point of Gateaux (Frchet) differentiability of the norm of X (m) + Xg(m)l [[f(m0)l[ + ll [[g[ess sup {[If(u) + kg(o)[[ e A 0} [[f(0)[[-6 whenever 0 [[ 2[--On the other hand, f + g[l >__ [[f[i [[ [[g][ > ]l f(0 )[[ ."flaeefore, lit + gil p (llt() + xm()ll e A O} wnr O< I1 .
o(g)(f) f()g()d() f el(,aO It follows from the results in Diestel and Uhl [i, p. 63], that if X has the Radon-NikodyCm property with respect to then o is a linear isometry of Lo(,X) onto B(LI(,O,X) Using this fact and Theorem 3.1 we get the following characterization of the points of Gateaux and Frchet dlfferentlabillty of the norm function on B(L I(,R0 X) THEOREM 4.1: Let X be a real Banach space and (fl,E,) a finite measure space such that X has the Radon-Nikodm property with respect to