COMPACTNESS IN SPACES OF LIPSCHITZ FUNCTIONS

. The aim of this paper is to prove a compactness criterium in spaces of Lipschitz and


INTRODUCTION
In the last years there have been an increasing interest in the study of Lipschitz functions and of spaces of Lipschitz functions, as a first step to extend to the nonlinear setting results from linear functional analysis.For instance, in the attempt to build a spectral theory for nonlinear operators, a special attention was paid to spectra of Lipschitz operators (see, e.g., [9], [2], [4]).Lipschitz duals, meaning spaces of Lipschitz functions on a metric linear space, were used to study best approximation problems in such spaces (see [10]).A good account on Banach spaces and Banach algebras of Lipschitz functions is given in the monograph [11].The monograph [6] contains a comprehensive study of Lipschitz functions on Banach spaces and their applications to the geometry of Banach spaces (e.g. the Lipschitz classification of Banach spaces).
As asserts Appell [1], apparently there is no compactness criterium in spaces of Hölder functions, and some criteria given in the literature turned to be false (e.g. that in [7]).The aim of this Note is to prove such a criterium (a true one, I hope) for families of Lipschitz and Fréchet differentiable mappings.The paper by J. Batt [5] contains a detailed study of compactness for nonlinear mappings and their adjoints, including Schauder type theorems.A Schauder type theorem for differentiable mappings was proved also by Yamamuro [12].

THE RESULT
Let X, Y be real or complex normed linear spaces, and Ω a subset of X. Denote by Lip(Ω, Y ) the space of all Lipschitz mappings from Ω to Y , i.e. those mappings f : Ω → Y for which the number (1) L(f is finite.The number L(f ) defined by ( 1) is called the Lipschitz norm of the mapping f , and it is the smallest Lipschitz constant for f .The function (The operations of addition and multiplication by scalars are defined pointwisely) If Ω is an open subset of X, denote by C 1 (Ω, Y ) the space of all continuously Fréchet differentiable mappings from Ω to Y , and for K ⊂ Ω put Let also L(X, Y ) denote the space of all continuous linear operators from X to Y equipped with the uniform norm.
The compactness result we shall prove is the following: Suppose that Z is a subset of C 1 Lip(K, Y ) such that (i) for every x ∈ K the set {f (x) : f ∈ Z} is totally bounded in L(X, Y ); (ii) for every x ∈ K and every > 0 there exists δ = δ(x, ) > 0 such that Then the set Z is totally bounded in Lip(K, Y ).Conversely, if the set Z ⊂ C 1 Lip(Ω, Y ) is totally bounded in Lip(Ω, Y ) then Z satisfies the conditions (i) and (ii).
As consequence, one obtains the following corollary.
Corollary 1.If Y is a Banach space and Z ⊂ C 1 Lip(K, Y ) is closed and satisfies the conditions (i) and (ii) from Theorem 1 then the set Z is compact in Lip(K, Y ).
The proof of Theorem 1 will be based on the following lemma: for every x in a neighborhood U ⊂ Ω of x 0 and g is Fréchet differentiable at x 0 , then Proof of Lemma 1. Suppose that g : Ω → Y satisfies (2).The differentiability of g at x 0 implies the existence of g (x 0 ) ∈ L(X, Y ) such that ( 6) where lim h→0 α(h) = 0.For n ∈ N choose δ n > 0 such that B(x 0 , δ n ) ⊂ Ω and Then, from (6), The inequality Conversely, suppose that g is Fréchet differentiable on an open convex neighborhood U ⊂ Ω of x 0 , and satisfies (4).
By the mean value theorem Suppose that the set Z ⊂ C 1 Lip(K, Y ) satisfies the conditions (i) and (ii), and let > 0 be given.By (ii), for every x ∈ K there exists δ x > 0 such that Since the set K is compact, there exists x 1 , . . ., x p in K such that as well as the set for k = 1, . . ., p.
We shall show that {f 1 , . . ., f n } is a 3 -net for the set Z with respect to the Lipschitz norm (1) on Lip(K, Y ).
By the mean value theorem, we have for x, y ∈ K Since ξ ∈ [x, y] ⊂ K, by (8) there exists k ∈ {1, . . ., p} such that ξ ∈ B(x k , δ k ).But then (12) (We have applied (7) to the first and the last term, and (10) to the middle one).