Rev. Anal. Num´er. Th´eor. Approx., vol. 30 (2001) no. 1, pp. 9–14 ictp.acad.ro/jnaat

COMPACTNESS IN SPACES OF LIPSCHITZ FUNCTIONS

S¸TEFAN COBZAS¸

Dedicated to the memory of Acad. Tiberiu Popoviciu

Abstract. The aim of this paper is to prove a compactness criterium in spaces of Lipschitz and Fr´echet differentiable mappings.

MSC 2000. 46E15.

1. 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 H¨older 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 Fr´echet 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].

“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, RO-3400 Cluj–Napoca, Romania, e-mail: [email protected].

2. 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) := sup{kf(x)−f(y)k/kx−yk:x, y∈Ω, x6=y}

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 L(·) is a seminorm on Lip(Ω, Y), so that (Lip(Ω, Y), L) is a seminormed space which is complete if Y is a Banach space. (The operations of addition and multiplication by scalars are defined pointwisely)

If Ω is an open subset ofX, denote byC^{1}(Ω, Y) the space of all continuously
Fr´echet differentiable mappings from Ω toY, and for K ⊂Ω put

C^{1}Lip(K, Y) :={f ∈Lip(K, Y) :∃F ∈C^{1}(Ω, Y) such that F|_{K}=f}.

Let also L(X, Y) denote the space of all continuous linear operators from X toY equipped with the uniform norm.

The compactness result we shall prove is the following:

Theorem 1. Let X, Y be normed spaces, Ω an open subset of X and K a compact convex subset of Ω.

Suppose that Z is a subset ofC^{1}Lip(K, Y) such that

(i) for every x∈K the set{f^{0}(x) :f ∈Z} is totally bounded in L(X, Y);

(ii) for everyx∈K and every >0 there existsδ =δ(x, )>0 such that

∀x^{0} ∈B(x, δ)⊂Ω, ∀f ∈Z kf^{0}(x)−f^{0}(x^{0})k ≤.
Then the set Z is totally bounded inLip(K, Y).

Conversely, if the setZ ⊂C^{1}Lip(Ω, Y)is totally bounded inLip(Ω, Y)then
Z satisfies the conditions (i) and (ii).

As consequence, one obtains the following corollary.

Corollary 1. IfY is a Banach space andZ ⊂C^{1}Lip(K, Y) is closed and
satisfies the conditions (i) and (ii) from Theorem 1then the set Z is compact
in Lip(K, Y).

The proof of Theorem 1 will be based on the following lemma:

Lemma 1. Let X, Y be normed spaces and Ω an open subset of X. If g: Ω→Y satisfies

(2) kg(x_{0})−g(x)k ≤λkx_{0}−xk,

for every x in a neighborhood U ⊂Ω of x0 and g is Fr´echet differentiable at
x_{0}, then

(3) kg^{0}(x_{0})k ≤λ.

Conversely, if g is Fr´echet differentiable on an open convex neighborhood U ⊂Ω of x0 and

(4) kg^{0}(x)k ≤λ, ∀x∈U,

then

(5) kg(x)−g(y)k ≤λkx−yk, ∀x, y∈U.

Proof of Lemma 1. Suppose that g: Ω → Y satisfies (2). The differentia-
bility ofg atx_{0} implies the existence ofg^{0}(x_{0})∈L(X, Y) such that

(6) g(x_{0}+h)−g(x_{0}) =g^{0}(x_{0})h+khkα(h),

where limh→0α(h) = 0. Forn∈Nchoose δn>0 such that B(x0, δn)⊂Ω and
kα(h)k ≤1/n, ∀h∈B(0, δ_{n}).

Then, from (6),

kg^{0}(x0)k ≤ kg(x_{0}+h)−g(x0)k+khkkα(h)k

≤(λ+ ^{1}_{n})khk.

The inequality

kg^{0}(x0)hk ≤(λ+_{n}^{1})khk, ∀h, khk ≤δn,
implies kg^{0}(x_{0})k ≤λ+ 1/n, ∀n∈N,so thatkg^{0}(x_{0})k ≤λ.

Conversely, suppose thatgis Fr´echet differentiable on an open convex neigh- borhoodU ⊂Ω of x0, and satisfies (4).

By the mean value theorem

kg(x)−g(y)k ≤ kx−yksup{kg^{0}(ξ)k:ξ∈[x, y]} ≤λkx−yk,
for all x, y∈U.

Lemma 1 is proved.

Proof of Theorem 1.

Suppose that the setZ ⊂C^{1}Lip(K, Y) satisfies the conditions (i) and (ii),
and let >0 be given.

By (ii), for every x∈K there existsδ_{x}>0 such that

(7) ∀f ∈Z and ∀x^{0} ∈B(x, δx)∩K kf^{0}(x)−f^{0}(x^{0})k ≤.
Since the set K is compact, there exists x_{1}, . . . , x_{p} inK such that

(8) K⊂

p

[

k=1

B(x_{k}, δ_{k}), where δ_{k}=δ_{x}_{k}.

By (i), the set Y_{k} = {f^{0}(x_{k}) : f ∈ Z} is totally bounded in L(X, Y), for
k= 1,2, . . . , p.It follows that the set

W =Y_{1}× · · · ×Y_{k}

is totally bounded in (L(X, Y))^{p} with respect to the norm
k(A_{1}, . . . , A_{p})k= max{kA_{1}k, . . . ,kA_{p}k},
as well as the set

H={(f^{0}(x_{1}), ..., f^{0}(x_{p})) :f ∈Z} ⊂W.

Therefore we can findf_{1}, . . . , f_{n} inZ such that

(9) ∀f ∈Z ∃j∈ {1, . . . , n} such that kf^{0}(xk)−f_{j}^{0}(xk)k ≤,
fork= 1, . . . , p.

We shall show that{f_{1}, . . . , f_{n}}is a 3-net for the setZ with respect to the
Lipschitz norm (1) on Lip(K, Y).

Letf ∈Z. By (9) there is j∈ {1, . . . , n} such that
(10) kf^{0}(x_{k})−f_{j}^{0}(x_{k})k ≤, for k= 1, . . . , n.

By the mean value theorem, we have for x, y∈K

(11) k(f −f_{j})(x)−(f −f_{j})(y)k ≤ kx−yksup{k(f^{0}−f_{j}^{0})(ξ)k:ξ ∈[x, y]}.

Since ξ ∈ [x, y] ⊂ K, by (8) there exists k ∈ {1, . . . , p} such that ξ ∈
B(x_{k}, δ_{k}). But then

(12) k(f^{0}−f_{j}^{0})(ξ)k ≤ kf^{0}(ξ)−f^{0}(xk)k+k(f^{0}−f_{j}^{0})(xk)k+kf_{j}^{0}(xk)−f_{j}^{0}(ξ)k ≤3.

(We have applied (7) to the first and the last term, and (10) to the middle one).

By (11) and (12)

k(f−f_{j})(x)−(f−f_{j})(y)k ≤3kx−yk
for all x, y∈K or, equivalently,

L(f −fj)≤3.

To prove the converse implication, suppose that Z ⊂Lip(Ω, Y)∩C^{1}(Ω, Y)
is totally bounded in Lip(Ω, Y), and let > 0 be given. Choose an -net
{f_{1}, . . . , f_{n}} ⊂Z, i.e. ∀f ∈Z∃j∈ {1, . . . , n}such that ∀x, y∈Ω:

k(f −fj)(x)−(f−fj)(y)k ≤kx−yk.

Taking into account Lemma 1, one obtains
kf^{0}(x)−f_{j}^{0}(x)k ≤,

for all x ∈Ω, showing that {f_{1}^{0}(x), . . . , f_{n}^{0}(x)} is an -net for the set{f^{0}(x) :
f ∈Z}. Therefore (i) holds.

To prove (ii), let > 0 and x ∈ Ω be fixed. Choose again an -net
{f_{1}, . . . , fn} for the set Z. Since the mappings fi are of class C^{1} there ex-
istsδ >0 such that

(13) ∀x^{0} ∈B(x, δ)⊂Ω and ∀i∈ {1, . . . , n} kf_{i}^{0}(x)−f_{i}^{0}(x^{0})k ≤.
Forf ∈Z choose j∈ {1, . . . , n} such that

(14) L(f −fj)≤.

By Lemma 1 this implies

(15) ∀y∈Ω k(f^{0}−f_{j}^{0})(y)k ≤.
Taking into account (13) and (15), we obtain

kf^{0}(x)−f^{0}(x^{0})k ≤ kf^{0}(x)−f_{j}^{0}(x)k+kf_{j}^{0}(x)−f_{j}^{0}(x^{0})k+kf_{j}^{0}(x^{0})−f^{0}(x^{0})k ≤3
for all x^{0} ∈B(x, δ), which shows that (ii) holds too.

Theorem 1 is completely proved.

REFERENCES

[1] J. Appell, The superposition operator in function spaces – A survey, Report No. 41, Universit¨at Augsburg 1987, 70 pp.

[2] J. AppellandM. D¨orfner,Some spectral theorem for nonlinear operators, Nonlinear Anal., 28(1997), 1955–1976.

[3] J. AppellandP. P. Zabrejko,Nonlinear Superposition Operators, Cambridge Univ.

Press, Cambridge, 1990.

[4] J. Appell, E. De Pascale and A. Vignoli, A comparison of different spectra for nonlinear operators, Nonlinear Anal.,40A(1999), 703–713.

[5] J. Batt, Nonlinear compact mappings and their adjoints, Math. Ann., 189 (1970), 5–25.

[6] Y. BenyaminiandJ. lindenstrauss,Geometric Nonlinear Functional Analysis, Col- loq. Publ. Vol. 49, AMS, Providence, RI, 2000.

[7] A. I. GusejnovandH. Sh. Muhtarov,Introduction to the Theory of Nonlinear Sin- gular Integral Equations, Nauka, Moskva, 1980 (in Russian).

[8] L. V. KantorovichandG. P. Akilov,Functional Analysis, Nauka, Moskva, 1984 (in Russian).

[9] I. J. MaddoxandA. W. Wickstead,The spectrum of uniformly Lipschitz mappings, Proc. Roy. Irish Acad.,89A(1989), 101–114.

[10] K. Schnatz, Nonlinear duality and best approximations in metric linear spaces, J.

Approx. Theory,49(1987), 201–218.

[11] N. Weaver,Lipschitz Algebras, World Scientific, Singapore, 1999.

[12] S. Yamamuro,The adjoint of differentiable mappings, J. Austral. Math. Soc.,8(1968), 397–409.

Received February 7, 2000.