Smooth extension of functions on a certain class of non-separable Banach spaces

0.21 MB·English

by  Mar Jimenez-Sevilla

SMOOTH EXTENSION OF FUNCTIONS ON A CERTAIN CLASS
OF NON-SEPARABLE BANACH SPACES
1
MAR JIME´NEZ-SEVILLA ANDLUIS SA´NCHEZ-GONZA´LEZ
1
0
2

n
a Abstract. LetusconsideraBanachspaceX withthepropertythateveryreal-
J
valued Lipschitz function f can be uniformly approximated by a Lipschitz, C1-
4 smooth function g with Lip(g)≤CLip(f) (with C depending only on the space
1
X). ThisisthecaseforaBanachspaceX bi-Lipschitzhomeomorphictoasubset

] ofc0(Γ),forsomesetΓ,suchthatthecoordinatefunctionsofthehomeomorphism
A are C1-smooth ([12]). Then, we prove that for every closed subspace Y ⊂X and
everyC1-smooth(Lipschitz)functionf :Y →R,thereisaC1-smooth(Lipschitz,
F
respectively) extension of f to X. We also study C1-smooth extensions of real-
.
h valuedfunctionsdefinedonclosedsubsetsofX. Theseresultsextendthosegiven
t
a in [5] to theclass of non-separable Banach spaces satisfying theabove property.
m
[ 1. Introduction and main results

2 In this note we consider the problem of the extension of a smooth function from
v
a subspace of an infinite-dimensional Banach space to a smooth function on the
7
4 whole space. More precisely, if X is an infinite-dimensional Banach space, Y is a
1 closed subspaceof X andf :Y → R is a Ck-smooth function, underwhatconditions
4 does there exist a Ck-smooth function F : X → R such that F = f? Under the
. |Y
2 assumption that Y is a complemented subspace of a Banach space, an extension of
0
a smooth function f : Y → R is easily found taking the function F(x) = f(P(x)),
0
1 where P : X → Y is a continuous linear projection. But this extension does not
: solve the problem since if a Banach space X is not isomorphic to a Hilbert space,
v
i then it has a closed subspace which is not complemented in X [15].
X
C. J. Atkin in [2] extends every smooth function f defined on a finite union of
r
a open convex sets in a separable Banach space which does not admit smooth bump
functions, provided that for every point in the domain of f, the restriction of f to a
suitable neighborhood of the point can be extended to the whole space. The most
fundamental result has been given by D. Azagra, R. Fry and L. Keener [5, 6]. They
have shown that if X is a Banach space with separable dual X∗, Y ⊂ X is a closed
subspace and f : Y → R is a C1-smooth function, then there exists a C1-smooth
extension F : X → R of f. They proved a similar result when Y is a closed convex
subset, f is defined on an open set U containing Y and f is C1-smooth on Y as
a function on X (i.e., f : U → R is differentiable at every point y ∈ Y and the
function Y 7→ X∗ defined as y 7→ f′(y) is continuous on Y). For a detailed account
Date: January 19, 2010.
2000 Mathematics Subject Classification. 46B20.
Key words and phrases. Smooth extensions; smooth approximations.
Supported in part by DGES (Spain) Project MTM2009-07848. L. S´anchez-Gonza´lez has also
been supported by grant MEC AP2007-00868.
1
2 MARJIME´NEZ-SEVILLAANDLUISSA´NCHEZ-GONZA´LEZ
of the related theory of (smooth) extensions to Rn of smooth functions defined in
closed subsets of Rn see [5]. Let us point out that the case of analytic maps is quite
different (see [1]).
The aim of this note is to extend the results in [5] to the general setting of
Banach spaces whereevery Lipschitz function can beapproximated by a C1-smooth,
Lipschitz function. By using the results of Lipschitz and smooth approximation of
Lipschitz mappings given by P. H´ajek and M. Johanis [11, 12] we shall extend the
results in [5] to a larger class of Banach spaces. We proceed along the same lines
as the proof of the separable case [5]. Additionally, we shall use the open coverings
given by M. E. Rudin, and the ideas of M. Moulis [16], P. H´ajek and M. Johanis
[12].
The notation we use is standard. In addition, we shall follow, whenever possible,
the notations given in [5] and [12]. We denote by ||·|| the norm considered in X and
by B(x,r) the open ball with center x ∈ X and radius r > 0 . If Y is a subspace of
X we denote the restriction of a function f : X → R to Y by f and we say that
|Y
F : X → R is an extension of f : Y → R if F = f. Recall that Lip(h) denotes
|Y
the Lipschitz constant of a Lipschitz function h : Y → R, where Y is a subset of a
Banach space X. We refer to [7] or [8] for any other definition.
Before stating the main results, let us define the property (∗) as the following
Lipschitz and C1-smooth approximation property for Lipschitz mappings.
Definition 1.1. A Banach space X satisfies property (∗) if there is a constant C ,
0
which only depends on the space X, such that, for any Lipschitz function f :X → R
and any ε > 0 there is a Lipschitz, C1-smooth function K : X → R such that
|f(x)−K(x)| < ε for all x ∈ X and Lip(K) ≤ C Lip(f).
0
We may equivalently say that X satisfies property (∗) if there is a constant C ,
0
which only depends on X, such that for any subset Y ⊂ X, any Lipschitz function
f : Y → R and any ε > 0 there is a C1-smooth, Lipschitz function K : X → R such
that
|f(y)−K(y)|< ε for all y ∈ Y and Lip(K)≤ C Lip(f).
0
Indeed, every real-valued Lipschitz function f defined on Y can be extended to
a Lipschitz function on X with the same Lipschitz constant (for instance F(x) =
inf {f(y)+Lip(f)||x−y||}).
y∈Y
Let us recall that every Banach space with separable dual satisfies property (∗)
[9, 12](see also [5]). J.M. Lasry and P.L. Lions proved in [14] that in a Hilbertspace
H and for every Lipschitz function f : H → R and every ε > 0 there exists a C1-
smoothandLipschitzfunctiong : H → Rsuchthat|f(x)−g(x)| < εforeveryx ∈ X
and Lip(g) = Lip(f) (see also [4]). Let us notice that the approximation in [14] is
for bounded functions. However it also works for unboundedLipschitz functions. P.
H´ajek and M. Johanis have proved in [11] that for any set Γ, c (Γ) satisfies property
0
(∗). Also, they give a sufficient condition for X to have property (∗): if there is a
bi-Lipschitz homeomorphism ϕ embedding X into c (Γ) whose coordinates e∗ ◦ϕ
0 γ
are C1-smooth, then X satisfies property (∗) [12]. More specifically, they showed
the following characterization: a Banach space X has property (∗) and is uniformly
homeomorphic to a subset of c (Γ) (for some set Γ) if and only if there is a bi-
0
Lipschitz homeomorphism ϕ embedding X into c (Γ) whose coordinates e∗ ◦ϕ are
0 γ
SMOOTH EXTENSION ON BANACH SPACES 3
C1-smooth. Let us note that there is a gap in the proof that WCG Banach spaces
with a C1-smooth and Lipschitz bump function satisfy property (∗) given in [10]
and it is unknown if the result holds.
We shall prove that, if X satisfies property (∗) and Y is a closed subspace of X,
thenforeveryC1-smoothreal-valued functionf definedonY,thereisaC1-extension
of f to X.
Theorem 1.2. Let X be a Banach space with property (∗). Let Y ⊂ X be a closed
subspace and f :Y → R a C1-smooth function. Then there is a C1-smooth extension
of f to X.
Furthermore, if the given C1-smooth function f is Lipschitz on Y, then there is
a C1-smooth and Lipschitz extension H : X → R of f to X such that Lip(H) ≤
CLip(f), where C is a constant depending only on X.
Corollary 1.3. Let M be a paracompact C1-smooth Banach manifold modeled on a
Banach space X with property (∗) (in particular, any Riemannian manifold), and
let N be a closed C1-smooth submanifold of M. Then, every C1-smooth function
f : N → R has a C1-smooth extension to M.
A similar result can bestated, as in the separable case [5], if Y is a closed convex
subset of X, f is defined on an open subset U of X such that Y ⊂ U and f : U → R
is C1-smooth on Y as a function on U, i.e. f :U → R is differentiable at every point
y ∈Y and the mapping Y 7→ X∗, y 7→ f′(y) is continuous on Y.
Theorem 1.4. Let X be a Banach space with property (∗), Y ⊂ X a closed convex
subset, U ⊂X an open set containing Y and f : U → R a C1-smooth function on Y
as a function on U. Then, there is a C1-smooth extension of f to X.
|Y
Furthermore, if the given C1-smooth function f is Lipschitz on Y, then there is
a C1-smooth and Lipschitz extension H : X → R of f to X such that Lip(H) ≤
|Y
CLip(f ), where C is a constant depending only on X.
|Y
Finally, we can conclude with the following corollary.
Corollary 1.5. LetX beaBanachspace suchthatthere isabi-Lipschitzhomeomor-
phism between X and a subset of c (Γ), for some set Γ, whose coordinate functions
0
are C1-smooth. Let Y ⊂ X be a closed subspace and f :Y → R a C1-smooth function
(respectively, C1-smooth and Lipschitz function). Then there is a C1-smooth exten-
sion H of f to X (respectively, a C1-smooth and Lipschitz extension H of f to X
with Lip(H)≤ CLip(f), where C is a constant depending only on X).
Corollary 1.6. Let X be one of the following Banach spaces:
(i) a Banach space such that there is a bi-Lipschitz homeomorphism between
X and a subset of c (Γ), for some set Γ, whose coordinate functions are
0
C1-smooth,
(ii) a Hilbert space.
Let Y ⊂ X be a closed convex subset, U ⊂ X an open set containing Y and f :
U → R be a C1-smooth function on Y as a function on U (respectively, C1-smooth
on Y as a function on U and Lipschitz on Y). Then, there is a C1-smooth extension
H of f to X (respectively, C1-smooth and Lipschitz extension H of f to X with
|Y |Y
Lip(H) ≤ CLip(f ), where C is a constant depending only on X).
|Y
4 MARJIME´NEZ-SEVILLAANDLUISSA´NCHEZ-GONZA´LEZ
In Appendix A, we study under what conditions a real-valued function defined
on a closed, non-convex subset Y of a Banach space X with property (∗) can be
extended to a C1-smooth function on X.
2. The proofs
The first result we shall need is the existence of C1-smooth and Lipschitz parti-
tions of unity on Banach spaces satisfying property (∗). Recall that a Banach space
X admits C1-smooth and Lipschitz partitions of unity when for every open cover
U = {U } ofX thereisacollectionofC1-smooth,Lipschitzfunctions{ψ } such
r r∈Ω i i∈I
that (1) ψ ≥ 0 on X for every i ∈ I, (2) the family {supp(ψ )} is locally finite,
i i i∈I
where supp(ψ ) = {x ∈ X :ψ (x) 6= 0}, (3) {ψ } is subordinated to U = {U } ,
i i i i∈I r r∈Ω
i.e. for each i ∈ I there is r ∈ Ω such that supp(ψ ) ⊂ U and (4) ψ (x) = 1
i r i∈I i
for every x ∈ X. Also let us denote by dist(A,B) the distance between two sets A
P
and B, that is to say inf{||a−b||: a ∈ A, b ∈ B}.
The following lemma gives us the tool to generalize the construction of suitable
open coverings on a Banach space, which will be the key to obtain a generalization
of the smooth extension result given in [5].
Lemma 2.1. (See M.E. Rudin [17]) Let E be a metric space, U = {U } be an
r r∈Ω
opencoveringofE. Then, thereareopenrefinements{Vn,r}n∈N,r∈Ω and{Wn,r}n∈N,r∈Ω
of U satisfying the following properties:
(i) V ⊂ W ⊂ U for all n ∈ N and r ∈ Ω,
n,r n,r r
(ii) dist(V ,E \W )≥ 1/2n+1 for all n ∈N and r ∈ Ω,
n,r n,r
(iii) dist(W ,W )≥ 1/2n+1 for any n ∈ N and r,r′ ∈ Ω, r 6= r′,
n,r n,r′
(iv) for every x ∈ E there is an open ball B(x,s ) of E and a natural number n
x x
such that
(a) if i> n , then B(x,s )∩W = ∅ for any r ∈ Ω,
x x i,r
(b) if i≤ n , then B(x,s )∩W 6= ∅ for at most one r ∈Ω.
x x i,r
P. H´ajek and M. Johanis [12] proved that if a Banach space X satisfies property
(*) then X admits C1-smooth and Lipschitz partitions of unity, which is, in turn,
equivalent to the existence of a σ-discrete basis B of the topology of X such that
for every B ∈ B there is a C1-smooth and Lipschitz function ψ : X → [0,1] with
B
B = ψ−1(0,∞) ([12], see also [13]). It is worth noting that given an open covering
{U } ofX, itisnotalways possibletoobtainaC1-smoothandLipschitzpartition
r r∈Ω
ofunity{ψ } withthesamesetofindexessuchthatsupp(ψ )⊂ U . Forexample,
r r∈Ω r r
if A is a non-empty, closed subset of X, W is an open subset of X such that A⊂ W
withdist(A,X\W) = 0, and{ψ ,ψ }isaC1-smoothpartitionofunitysubordinated
1 2
to {W,X \A}, then ψ (A) = 1 and ψ (X \W) = 0 and thus ψ is not Lipschitz.
1 1 1
Nevertheless, in order to prove Theorem 2.4 we only need the following statement.
Lemma 2.2. Let X be a Banach space with property (∗). Then, for every {U }
r r∈Ω
open covering of X, there is an open refinement {Wn,r}n∈N,r∈Ω of {Ur}r∈Ω satisfying
the properties of Lemma 2.1, and there is a Lipschitz and C1-smooth partition of
unity {ψn,r}n∈N,r∈Ω such that supp(ψn,r) ⊂ Wn,r ⊂ Ur and Lip(ψn,r) ≤ C025(2n−1)
for every n ∈ N and r ∈ Ω.
Proof. Letusconsideranopencovering{U } ofX. ByLemma2.1,thereareopen
r r∈Ω
refinements {Vn,r}n∈N,r∈Ω and {Wn,r}n∈N,r∈Ω of {Ur}r∈Ω satisfying the properties
SMOOTH EXTENSION ON BANACH SPACES 5
(i)-(iv)ofLemma2.1. ConsiderthedistancefunctionD (x) = dist(x,X\ W )
n r∈Ω n,r
which is 1-Lipschitz. By applying property (∗), there is a C1-smooth, C -Lipschitz
0
S
function g : X → R such that |g (x) − D (x)| < 1 for every x ∈ X. Thus
n n n 2n+3
g (x) > 1 wheneverx ∈ V andg (x) < 1 wheneverx ∈ X\ W .
n 2n+2 r∈Ω n,r n 2n+3 r∈Ω n,r
Bycomposingg withasuitableC∞-smoothfunctionϕ : R → [0,1]withLip(ϕ ) ≤
n n n
2n+4 we obtain a C1-smootSh function h := ϕ (g ) that is zero on aSn open set
n n n
including X\ W , h ≡ 1 and Lip(h ) ≤ C 2n+4. Now, let us define
r∈Ω n,r n|Sr∈ΩVn,r n 0
HS1 = h1, and Hn = hn(1−h1)···(1−hn−1) for n ≥ 2.
It is clear that H (x) = 1 for all x ∈ X. Since supp(h ) ⊂ W and
n n n r∈Ω n,r
W ∩W = ∅ for every n ∈ N and r 6= r′, we can write h = h , where
n,r n,r′ P n Sr∈Ω n,r
h (x) = h (x) on W and supp(h ) ⊂ W . Notice that Lip(h ) ≤ Lip(h ) ≤
n,r n n,r n,r n,r n,r n
C 2n+4. Let us define, for every r ∈ Ω, P
0
ψ = h , and ψ = h (1−h )···(1−h ) for each n ≥2.
1,r 1,r n,r n,r 1 n−1
The functions {ψn,r}n∈N,r∈Ω satisfy that
(i) they areC1-smoothandLipschitz, withLip(ψ )≤ C n+42i = C 25(2n−
n,r 0 i=5 0
1),
P
(ii) supp(ψ )⊂ supp(h )⊂ W , and
n,r n,r n,r
(iii) for every x ∈ X,
n−1
ψ (x)= ψ (x)+ h (x) (1−h (x)) = H (x) = 1.
n,r 1,r n,r i n
n∈N,r∈Ω r∈Ω n≥2 r∈Ω ! i=1 n∈N
X X X X Y X
(cid:3)
An alternative proof of Theorem 2.4 uses the existence of the σ-discrete basis for
X previously mentioned and the construction, as above, of suitable C1-smooth and
Lipschitz partitions of unity subordinated to any subfamily of this basis.
Thefollowinglemmaisanecessarymodificationofproperty(∗)toshowthemain
results.
Lemma 2.3. Let X be a Banach space with the property (∗). Then for every subset
Y ⊂ X, every continuous function F : X → R such that F is Lipschitz, and every
|Y
ε> 0, there exists a C1-smooth function G : X → R such that
(i) |F(x)−G(x)| ≤ ε for all x ∈ X, and
(ii) Lip(G ) ≤ C Lip(F ). Moreover, ||G′(y)|| ≤ C Lip(F ) for all y ∈ Y,
|Y 0 |Y X∗ 0 |Y
where C is the constant given by property (∗).
0
(iii) In addition, if F is Lipschitz on X, there exists a constant C ≥ C that
1 0
depends only on X, such that the function G can be chosen to be Lipschitz
on X and Lip(G) ≤ C Lip(F).
1
Proof. AssumethatthefunctionF : X → RiscontinuousonX andF isLipschitz.
|Y
Since X admits C1-smooth partitions of unity, there is (by [7, Theorem VIII 3.2])
a C1-smooth function h : X → R, such that |F(x)−h(x)| < ε for all x ∈ X. Let
F : X → R be a Lipschitz extension of F to X with Lip(F) = Lip(F ). Let us
|Y |Y
apply property (∗) to F to obtain a C1-smooth, Lipschitz function g : X → R such
e e
that |F(x)−g(x)| < ε/4 for every x ∈ X, and Lip(g) ≤ C Lip(F ). Consider the
0 |Y
e
e
6 MARJIME´NEZ-SEVILLAANDLUISSA´NCHEZ-GONZA´LEZ
open sets A = {x ∈ X : |F(x)−F(x)| < ε/4}, B = {x ∈ X : |F(x)−F(x)| < ε/2}
in X and the closed set C = {x ∈ X : |F(x)−F(x)| ≤ ε/4} in X. Then Y ⊂ A ⊂
C ⊂ B. By [7, Proposition VIIIe3.7] there is a C1-smooth function ue: X → [0,1]
such that e
1 if x ∈C,
u(x) =
(0 if x ∈X \B.
Let us define G: X → R as
G(x) := u(x)g(x)+(1−u(x))h(x).
It is clear that G is a C1-smooth function. Since u(x) = 0 for all x ∈ X \B, we
deduce that |F(x)−G(x)| = |F(x)−h(x)| < ε for all x ∈ X \B. Now, if x ∈ B,
then |F(x)−G(x)| ≤ u(x)|F(x)−g(x)|+(1−u(x))|F(x)−h(x)| ≤ u(x)(|F(x)−
F(x)|+|F(x)−g(x)|)+(1−u(x))|F(x)−h(x)| ≤ ε. Finally, since u(x) = 1 and
G(x) = g(x) for every x ∈ A, we obtain that Lip(G ) = Lip(g ) ≤ C Lip(F )
|Y |Y 0 |Y
aend ||G′(ye)||X∗ = ||g′(y)||X∗ ≤ C0Lip(F|Y) for all y ∈ Y.
Let us now assume that F is Lipschitz on X. Let us apply property (∗) to F and
F (where F is a Lipschitz extension of F to X with Lip(F) = Lip(F )). Thus,
|Y |Y
we obtain C1-smooth and Lipschitz functions g, h : X → R such that
e e e
(a) |F(x)−g(x)| < ε/4, for all x ∈X,
(b) |F(x)−h(x)| < ε, for every x ∈ X, and
(c) Leip(g) ≤ C Lip(F ) and Lip(h) ≤ C Lip(F).
0 |Y 0
WetakeagainthesubsetsA, B andC asinthepreviouscase. Noticethatdist(C,X\
B) ≥ ε = ε′. Let us prove that there is a C1-smooth, Lipschitz
4(Lip(F)+Lip(F ))
|Y
function u : X → [0,1] such that u(x) = 1 on C and u(x) = 0 on X \ B, with
Lip(u) ≤ 9C0(Lip(F)+Lip(F|Y)). Let us consider 0 < r ≤ ε′/4, and the distance
ε
function D : X → R, D(x) = dist(x,C). Since the function D is 1-Lipschitz, we
apply property (∗) to obtain a C1-smooth, Lipschitz function R : X → R such that
Lip(R) ≤ C and |D(x)−R(x)| < r for all x ∈ X. Also, let us take a C1-smooth and
0
Lipschitz function ϕ : R → [0,1] with (i) ϕ(t) = 1 whenever |t| ≤ r, (ii) ϕ(t) = 0
whenever|t|≥ ε′−r and(iii)Lip(ϕ) ≤ 9 ≤ 9 . Next, wedefinetheC1-smooth
8(ε′−2r) 4ε′
function u:X → [0,1], u(x) = ϕ(R(x)). Notice that Lip(u) ≤ 9C0(Lip(F)+Lip(F|Y)).
ε
Let us now consider G: X → R as
G(x) = u(x)g(x)+(1−u(x))h(x).
Clearly G is C1-smooth on X. We follow the above proof to obtain that |F(x) −
G(x)| < ε on X, Lip(G ) = Lip(g ) ≤ C Lip(F ) and ||G′(y)|| ≤ C Lip(F )
|Y |Y 0 |Y X∗ 0 |Y
for all y ∈ Y. Additionally, if x ∈ X \ B, then u(x) = 0, G(x) = h(x), and
SMOOTH EXTENSION ON BANACH SPACES 7
||G′(x)|| = ||h′(x)|| ≤ C Lip(F). For x ∈ B, we have
X∗ X∗ 0
||G′(x)|| ≤
X∗
≤ ||g(x)u′(x)+h(x)(1−u)′(x)|| +||u(x)g′(x)+(1−u(x))h′(x)|| ≤
X∗ X∗
≤ ||(g(x)−F(x))u′(x)+(h(x)−F(x))(1−u)′(x)|| +C Lip(F) ≤
X∗ 0
≤ (|g(x)−F(x)|+|F(x)−F(x)|+|h(x)−F(x)|)||u′(x)|| +C Lip(F) ≤
X∗ 0
≤ 7ε · 9C0(eLip(F)+eLip(F|Y)) +C0Lip(F) ≤ 33C0Lip(F).
4 ε
We define C := 33C and obtain that Lip(G) ≤ C Lip(F).
1 0 1
(cid:3)
The following approximation result is the key to prove Theorem 1.2. Recall that
the separable case was given in [5, Theorem 1].
Theorem 2.4. Let X be a Banach space with property (∗), and Y ⊂ X a closed
subspace. Let f : Y → R be a C1-smooth function, and F a continuous extension of
f to X. Then, for every ε > 0 there exists a C1-smooth function G : X −→ R such
that if g =G then
|Y
(i) |F(x)−G(x)| < ε on X, and
(ii) kf′(y)−g′(y)k < ε on Y.
Y∗
(iii) Furthermore, if f is Lipschitz on Y and F is a Lipschitz extension of f to
X, then the function G can be chosen to be Lipschitz on X and Lip(G) ≤
C Lip(F), where C is a constant only depending on X.
2 2
Proof. Wefollowthestepsgivenintheproofoftheseparablecasewiththenecessary
modifications. Notice that by the Tietze theorem, a continuous extension F of f
always exists. Since X is a Banach space, Y ⊂ X is a closed subspace and f′ is a
continuous function on Y, there exists {B(y ,r )} a covering of Y by open balls
γ γ γ∈Γ
of X, with centers y ∈ Y, 0 6∈ Γ, such that
γ
ε
(2.1) kf′(y )−f′(y)k <
γ Y∗
8C
0
on B(y ,r )∩Y, where C is the positive constant given by property (∗) (which
γ γ 0
depends only on X). We denote B := B(y ,r ).
γ γ γ
LetusdefineT anextension of thefirstorderTaylor Polynomial of f aty given
γ γ
by T (x) = f(y )+H(f′(y ))(x−y ), for x ∈ X, where H(f′(y )) ∈ X∗ denotes
γ γ γ γ γ
a Hahn-Banach extension of f′(y ) with the same norm, i.e. ||H(f′(y ))|| =
γ γ X∗
||f′(y )|| . Notice that T satisfies the following properties:
γ Y∗ γ
(B.1) T is C∞-smooth on X,
γ
(B.2) T′(x) = H(f′(y )) for all x ∈X, T′(y) | = f′(y ) for every y ∈ Y, and
γ γ γ Y γ
(B.3) from (B.2), (2.1) and the fact that B ∩Y is convex, we deduce Lip((T −
γ γ
F)| )≤ ε .
Bγ∩Y 8C0
Since F : X → R is a continuous function, and X admits C1-smooth partitions
of unity, there is a C1-smooth function F :X → R such that |F(x)−F (x)| < ε for
0 0 2
every x ∈ X.
Let us denote B := X \ Y, Σ := Γ ∪ {0}, and C := {B : β ∈ Σ}, which
0 β
is a covering of X. By Lemma 2.2, there is an open refinement {Wn,β}n∈N,β∈Σ
8 MARJIME´NEZ-SEVILLAANDLUISSA´NCHEZ-GONZA´LEZ
of C = {B : β ∈ Σ} satisfying the four properties of Lemma 2.1. In particular,
β
W ⊂ B and for each x ∈ X there is an open ball B(x,s ) of X with center x
n,β β x
and radius s >0, and a natural number n such that
x x
(1) if i> n , then B(x,s )∩W = ∅ for every β ∈ Σ,
x x i,β
(2) if i≤ n , then B(x,s )∩W 6= ∅ for at most one β ∈ Σ.
x x i,β
Thereis, by Lemma 2.2, aC1-smooth and Lipschitzpartition of unity {ψn,β}n∈N,β∈Σ
such that supp(ψ )⊂ W ⊂ B and Lip(ψ ) ≤ C 25(2n −1) for every (n,β) ∈
n,β n,β β n,β 0
N×Σ.
Let us define L := max{Lip(ψ ),1} for every n ∈ N and β ∈ Σ. Now, for
n,β n,β
every n ∈ N and γ ∈ Γ, we apply Lemma 2.3 to T − F on B ∩ Y to obtain a
γ γ
C1-smooth map δ : X −→ R so that
n,γ
ε
(C.1) |T (x)−F(x)−δ (x)| < for every x ∈ X
γ n,γ 2n+2L
n,γ
and
ε
(C.2) kδ′ (y)k ≤ for every y ∈ B ∩Y.
n,γ X∗ 8 γ
From inequality (2.1), (B.2) and (C.2), we have for all y ∈ B ∩Y,
γ
ε
kT′(y)−f′(y)−δ′ (y)k ≤ kT′(y)−f′(y)k +kδ′ (y)k < .
γ n,γ Y∗ γ Y∗ n,γ Y∗ 4
Notice that in the above inequality we consider the norm on Y∗ (i.e. the norm of
the functional restricted to Y). Let us define
F (x) if β = 0,
(2.2) ∆n(x) = 0
β
(Tβ(x)−δn,β(x) if β ∈ Γ.
Thus, |∆n(x)−F(x)| < ε whenever n ∈ N, β ∈ Σ and x ∈ B . We now define
β 2 β
G(x) = ψ (x)∆n(x).
n,β β
(n,β)∈N×Σ
X
Since {ψn,β}n∈N,β∈Σ is locally finitely nonzero, then G is C1-smooth. Now, if x ∈ X
and ψ (x) 6= 0, then x ∈ B and thus
n,β β
ε
|G(x)−F(x)| ≤ ψ (x)|∆n(x)−F(x)| ≤ ψ (x) < ε.
n,β β n,β 2
(n,β)∈N×Σ (n,β)∈N×Σ
X X
Let us now estimate the distance between the derivatives. From the definitions
given above, notice that:
(D.1) Since ψ (x) = 1 for all x ∈ X, we have that ψ′ (x) = 0 for
N×Σ n,β N×Σ n,β
all x ∈ X.
P P
(D.2) Thus, we can write f′(y) = (ψ′ (y)) f(y)+ ψ (y)f′(y), for
N×Σ n,β |Y N×Σ n,β
every y ∈Y.
P P
(D.3) supp(ψ ) ⊂ B = X \Y, for all n.
n,0 0
(D.4) G′(x) = ψ′ (x)∆n(x)+ ψ (x)(∆n)′(x), for all x ∈ X.
N×Σ n,β β N×Σ n,β β
(D.5) If g = G then g′(y) = ψ′ (y) ∆n(y)+ ψ (y)(∆n)′(y) ,
P|Y N×ΣPn,β |Y β N×Σ n,β β |Y
for every y ∈ Y.
P P
SMOOTH EXTENSION ON BANACH SPACES 9
(D.6) Properties (1) and (2) of the open refinement {W } imply that for every
n,β
x ∈ X and n ∈ N, there is at most one β ∈ Σ, which we shall denote by
β (n),suchthatx ∈ supp(ψ ). Inthecasethaty ∈ Y,thenβ (n)∈ Γ. We
x n,β y
define F := {(n,β) ∈ N×Σ : x ∈ supp(ψ )}. In particular, F ⊂ N×Γ,
x n,β y
whenever y ∈Y.
We obtain, for y ∈ Y,
(2.3)
kg′(y)−f′(y)k ≤
Y∗
≤ kψ′ (y)k |T (y)−f(y)−δ (y)|
n,β Y∗ β n,β
(n,Xβ)∈Fy
+ ψ (y)kT′(y)−f′(y)−δ′ (y)k ≤
n,β β n,β Y∗
(n,Xβ)∈Fy
≤ L |T (y)−f(y)−δ (y)|+
n,βy(n) βy(n) n,βy(n)
{n:(n,βXy(n))∈Fy}
+ ψ (y)kT′ (y)−f′(y)−δ′ (y)k ≤
n,βy(n) βy(n) n,βy(n) Y∗
{n:(n,βXy(n))∈Fy}
ε ε ε ε
≤ (L +ψ (y) ) ≤ + < ε,
n,βy(n) 2n+2L n,βy(n) 4 4 4
{n:(n,βXy(n))∈Fy} n,βy(n)
where all the functionals involved are considered restricted to Y.
Let us now consider the case when f is C1-smooth and Lipschitz on Y and F
is a Lipschitz extension of f on X. In this case, we can assume that f is not
constant (otherwise the assertion is trivial) and thus Lip(F) ≥ Lip(f) > 0. Let
us fix 0 < ε < Lip(F). If we follow the above construction for the open covering
{B } ofX satisfyingtheconditions (2.1), (B.1), (B.2) and(B.3), weadditionally
β β∈Σ
obtain
(B.4) T −F is Lipschitz on X and Lip(T −F) ≤ Lip(f)+Lip(F) forevery β ∈ Γ.
β β
Also, the construction of the open refinement {Wn,β}n∈N,β∈Σ of C = {Bβ}β∈Σ, the
Lipschitz partition of unity {ψn,β}n∈N,β∈Σ and the definition of Ln,β are similar to
the previous case (recall that Σ = Γ∪{0} and B = X \Y).
0
Now, for any n ∈ N and β ∈ Γ, we apply Lemma 2.3 to T −F on B ∩Y to
β β
obtain a C1-smooth map δ : X −→ R satisfying conditions (C.1) , (C.2) and
n,β
(C.3) Lip(δ ) ≤ C Lip(T −F)≤ C (Lip(f)+Lip(F)).
n,β 1 β 1
Besides, for all n ∈ N and β = 0, by applying property (∗), we select a C1-smooth
function Fn :X → R, such that
0
ε
|Fn(x)−F(x)| < for every x ∈X and Lip(Fn) ≤ C Lip(F).
0 2n+2L 0 0
n,0
Thus, if we define ∆n : X → R
β
Fn(x) if β = 0,
(2.4) ∆n(x) = 0
β
(Tβ(x)−δn,β(x) if β ∈ Γ,
10 MARJIME´NEZ-SEVILLAANDLUISSA´NCHEZ-GONZA´LEZ
we obtain for every β ∈ Σ,
ε
|∆n(x)−F(x)| < whenever x ∈X
β 2n+2L
n,β
and Lip(∆n) ≤ max{(1 + C )Lip(f) + C Lip(F),C Lip(F)} ≤ RLip(F) where
β 1 1 0
R := 1+2C is a constant depending only on X. Similarly to the first case, the
1
definition of G is
G(x) = ψ (x)∆n(x).
n,β β
(n,β)∈N×Σ
X
The proofs that G is C1-smooth, |G(x) − F(x)| < ε for all x ∈ X and ||g′(y) −
f′(y)|| < ε for all y ∈ Y (recall that g = G ) follow along the same lines.
Y∗ |Y
In addition, let us check that G is Lipschitz. From properties (D.1) to (D.6), in
particular from the fact that ψ′ (x) = 0, we deduce that
(n,β)∈Fx n,β
(2.5) P
||G′(x)|| ≤ ||ψ′ (x)|| |∆n(x)−F(x)|+ ψ (x)||(∆n)′(x)|| ≤
X∗ n,β X∗ β n,β β X∗
(n,Xβ)∈Fx (n,Xβ)∈Fx
ε
≤ L + ψ (x)RLip(F) ≤
n,β(n) 2n+2L n,β(n)
n,β(n)
{n:(n,βX(n))∈Fx} {n:(n,βX(n))∈Fx}
ε
≤ +RLip(F).
4
Since ε < Lip(F), then Lip(G) ≤ C Lip(F) where C := R+ 1 and this finishes the
2 2 4
proof. (cid:3)
The above theorem provides the tool to prove the extension Theorem 1.2, which
states that every C1-smooth function (C1-smooth and Lipschitz function) defined on
a closed subspace has a C1-smooth extension (C1-smooth and Lipschitz extension,
respectively) to X.
Proof of Theorem 1.2. For every bounded,Lipschitz function, h :Y → R, wedefine
h(x) := min{||h|| ,inf {h(y) +Lip(h)||x − y||}} for any x ∈ X. The function
∞ y∈Y
h is a Lipschitz extension of h to X with Lip(h) = Lip(h) and ||h|| = ||h|| =
∞ ∞
sup{|h(y)| :y ∈ Y}.
Letusassumethatthefunctionf :Y → RisC1-smoothandconsiderF :X → R
a continuous extension of f to X and ε > 0. We apply Theorem 2.4 to deduce the
existence of a C1-smooth function G : X → R such that if g := G , we have
1 1 1|Y
(i) |F(x)−G (x)| < ε/2 for x ∈ X, and
1
(ii) ||f′(y)−g′(y)|| < ε/2C for y ∈ Y. Since Y is convex, then we have
1 Y∗ 2
Lip(f −g ) ≤ ε/2C .
1 2
The function f −g is bounded by ε/2 and ε -Lipschitz on Y. Thus, there exists
1 2C2
a bounded, Lipschitz extension to X, f −g , satisfying |(f −g )(x)| ≤ ε/2 on X
1 1
and Lip(f −g ) ≤ ε/2C . Following the construction given for the separable case
1 2
[5], we apply Theorem 2.4 (Lipschitz case) to f −g to obtain a C1-smooth function
1
G :X → R such that if g := G , we have
2 2 2|Y
(i) |(f −g )(x)−G (x)| < ε/22 for x ∈X,
1 2
(ii) ||f′(y)−(g′(y)+g′(y))|| < ε/22C for y ∈Y, and
1 2 Y∗ 2
(iii) Lip(G ) ≤ C Lip(f −g ) ≤ ε/2.
2 2 1