Smooth extensions of functions on separable Banach spaces

Smooth extensions of functions on separable Banach spaces

D. Azagra, R. Fry, and L. Keener

Abstract. Let X be a Banach space with a separable dual X . Let Y X be a closed subspace, and f : Y ! R a C¹-smooth function.

Then we show there is a C¹ extension of f to X.

1. Introduction

In this note we address the problem of the extension of smooth functions from subsets of Banach spaces to smooth functions on the whole space.

For our results, smoothness is meant in the Fréchet sense, and we shall restrict our attention to real-valued functions. To state the problem more precisely, given a Banach space X, a closed subset Y; and a C^p-smooth function f : Y ! R; when is it possible to …nd a C^p-smooth map F : X ! R such that F jY= f ?

We should note that when Y is a complemented subspace of an arbitrary Banach space X, the extension problem can be easily solved. Indeed, let P : X ! Y be a continuous linear projection, and f : Y ! R a C^p- smooth function. Then F (x) = f (P x) de…nes a C¹ extension of f to X.

Unfortunately, not every closed subspace Y of a separable Banach space X is complemented. In fact, a classic result of Lindenstrauss and Tzafriri [LT] states that the only Banach space all of whose closed subspaces are complemented is (up to renorming) a Hilbert space, so this trick only works when X is a Hilbert space.

When p = 0; this question is the problem of the continuous extension of functions from closed subsets. A complete characterization was given by the well known theorem of Tietze (see e.g., [Wi]) which we recall states that X is a normal space i¤ for every closed subset Y X and continuous function f : Y ! R; there exists a continuous extension F : X ! R of f:

Such characterizations in the di¤erentiable case, where p 1; are more delicate. When X = R; Y X is a subset, and p 1; necessary and

1991 Mathematics Subject Classi…cation. Primary 46B20.

Key words and phrases. Smooth extension, Banach space. The second named author is partly supported by NSERC (Canada).

1

su¢ cient conditions (in terms of divided di¤erences) for the existence of C^p-extensions to R of C^p functions on Y were given by H. Whitney [W1], [W2]. Apparently, Whitney intended to …nd such a characterization in the case X = Rⁿ with n > 1; but a sequel to the paper [W2] never appeared.

Major advances in this area occurred some twenty years later with the fundamental work of Glaeser [G] who solved the problem when n 1 and p = 1. Subsequent work included that of Brudnyi and Shvartsman [BS1], [BS2], Bierstone, Milman and Pawlucka [BMP1], [BMP2], and in partic- ular the striking results of C. Fe¤erman [Fe1], [Fe2], [Fe3]. For example, in [Fe2] a complete characterization is given of when a real-valued function de…ned on a compact subset of Rⁿ is the restriction of a C^m-smooth map on Rⁿ:

In this paper we consider the case when X is a separable Banach space which admits a C¹-smooth norm, a condition which is well known to be equivalent to X being separable [DGZ]. Then if Y X is a closed subspace and f : Y ! R is C¹-smooth, we show there exists a C¹ extension F : X ! R. If we require only that Y X be closed and not necessarily a subspace, then a similar conclusion holds under the stronger assumption that f is de…ned on a neighbourhood U Y and is C¹-smooth on Y as a function on X (i.e., f⁰(y) 2 X for y 2 Y and y ! f⁰(y) is continuous).

We observe, however, that in general the smooth extension problem has a negative solution. We give here three examples.

(1) In [Z] (see also [DGZ, Theorem II.8.3, page 82]) an example is given of a separable Banach space Y X = C [0; 1] ; and a Gâteaux smooth norm on Y that cannot be extended to a Gâteaux smooth norm on X:

(2) Also in [Z], it is shown that for 1 < p < 2; there is a subspace Y L_p isomorphic to Hilbert space, such that the Hilbertian norm of Y cannot be extended to a function ' on Lp which is Fréchet smooth on the unit sphere S_Y of Y as a function on X with y ! '⁰(y) locally Lipschitz from S_Y to X : Since every C² smooth function has a locally Lipschitz derivative, this immedi- ately shows that, given a C¹ smooth function f on Y L_p (with 1 < p < 2), in general there is no C² smooth extension of f to L_p. (3) As suggested to us by R. Aron [A] (see also Example 2.1 [AB]).

Let X = C [0; 1] which has the Dunford-Pettis Property, and hence the polynomial Dunford-Pettis Property (i.e., if P : X ! R is a polynomial and x_i ! 0; then P (x^w i) ! P (0)). Now Y = l2 X by the Banach-Mazur Theorem, and we consider f (x) = kxk²l2: If f extended to a C²-smooth function F on an open neighbourhood of l₂ in X; then

P (h) = (1=2) F⁰⁰(0) (h; h)

would be a polynomial on X: But ei

! 0 in lw ² and so in X; and thus as noted above we would have 1 = P (e_j) ! P (0) = 0; a contradiction.

One can compare the results of this note with the work of C.J. Atkin [At]. The programme of Atkin is to …nd smooth extension results for smooth functions f de…ned on …nite unions of open, convex sets in separable Banach spaces X that do not admit C^p-smooth norms, or even C^p-smooth bump functions. In order to achieve this, however, it is assumed in [At] that the function f already possesses smooth extensions to all of X in a neighbour- hood of every point in its domain. Finally we mention the result [DGZ, Proposition VIII.3.8], which states, in particular, that for weakly compactly generated X which admit C^p-smooth bump functions, for any closed sub- set Y X and continuous function f : Y ! R; there exists a continuous extension of f to X which is C^p-smooth on XnY:

We remark that the situation for analytic maps is quite di¤erent. Indeed, the paper by R. Aron and P. Berner [AB] characterizes the existence of analytic extensions from subspaces in terms of the existence of a linear extension operator. In particular, in the real case they prove, among many other equivalences, that if Y is a closed subspace of a Banach space X; the the following are equivalent (recall that Z is a C-space if it is complemented in its second dual Z ):

(1) For any C-space Z, and (real) analytic map f : Y ! Z which is bounded on bounded sets, there exists an analytic extension F : X ! Z of f; also bounded on bounded sets.

(2) There exists a continuous, linear extension operator T : Y ! X : We have combined some recent work on smooth approximation of Lip- schitz mappings [F] with some techniques of Moulis [M], the Bartle-Graves selector theorem, and the classical method of Tietze to deduce our principal results.

Our notation is standard, with X typically denoting a (real) Banach space. We shall denote an open ball with centre x 2 X and radius r > 0 either by B_r(x) ; B (x; r) ; or B_r if the centre is understood. We write the closed unit ball of a Banach space X as B_X: If Y X; we denote the restriction of a function f : X ! R to Y by f jY; and we say that a map K : X ! R is an extension of f : Y ! R if K (y) = f (y) for all y 2 Y:

We denote the Fréchet derivative of a function g at x in the direction h by g⁰(x) (h) : As noted above, if Y X and U Y is open, we say that f : U ! R is C¹-smooth on Y as a function on X if f⁰(y) 2 X and y ! f⁰(y) is continuous for y 2 Y: For any unde…ned terms we refer the reader to [FHHMPZ], [DGZ].

2. Main Results

An essential tool shall be the following easy consequence of the main theorem in [F], see also [HJ].

Lemma 1. Let X be a separable Banach space which admits a C¹-smooth norm. Then there exists a constant C₀ 1 such that, for every subset Y X, every -Lipschitz function f : Y ! R, and every " > 0, there exists a C₀ -Lipschitz, C¹-smooth function K : X ! R such that for all y 2 Y;

jf(y) K(y)j < ":

Proof. We note that the proof of the main result in [F] shows that there is C 1 such that for every 1-Lipschitz function g : X ! [0; 10], there exists a C¹ function ' : X ! R such that

(1) jg(x) '(x)j 1=8 for all x 2 X (2) ' is C-Lipschitz.

We …rst see that this result remains true for functions g taking values in R if we replace 1=8 with 1 and we allow C to be slightly larger. Indeed, by considering the function h = ', where is a C¹ smooth function : R ! [0; 10] such that jt (t)j 1=4 if t 2 [0; 10], (t) = 0 for t 1=8, and (t) = 10 for t 10 1=8, we get the following result: there exists C₀:= C Lip( ) such that for every 1-Lipschitz function g : X ! [0; 10] there exists a C¹ function h : X ! [0; 10] such that

(1) jg(x) h(x)j 1=2 for all x 2 X (2) h is C₀-Lipschitz

(3) g(x) = 0 =) h(x) = 0, and g(y) = 10 =) h(y) = 10.

Now, for a 1-Lipschitz function g : X ! [0; +1) we can write g(x) = P₁

n=0gn(x), where gn(x) =

8>

<

>:

g(x) 10n if 10n g(x) 10(n + 1);

0 if g(x) 10n;

10 if 10(n + 1) g(x)

and the sum is locally …nite. The functions g_n are clearly 1-Lipschitz and take values in the interval [0; 10], so there are C¹ functions hn: X ! [0; 10]

such that for all n 2 N we have that hn is C₀-Lipschitz, jgn h_nj 1=2, and hn is 0 or 10 wherever gn is 0 or 10. It is easy to check that the function h : X ! [0; +1) de…ned by h =P₁

n=0h_n is C¹ smooth, C₀-Lipschitz, and satis…es jg hj 1. This argument shows that there is C₀ 1 such that for any 1-Lipschitz function g : X ! [0; +1), there exists a C¹ function h : X ! [0; +1) such that

(1) jg(x) h(x)j 1 for all x 2 X (2) h is C0-Lipschitz

(3) g(x) = 0 =) h(x) = 0.

Finally, for an arbitrary 1-Lipschitz function g : X ! R, we can write g = g⁺ g and apply this result to …nd C¹ smooth, C₀-lipschitz functions

h⁺; h : X ! [0; +1) so that h := h⁺ h is C₀-Lipschitz, C¹ smooth, and satis…es jg hj 1.

Now let us prove the Lemma. By replacing f with the function x 7! inf

y2Yff(y) + kx ykg;

which is an -Lipschitz extension of f to X, we may assume that Y = X. Consider the function g : X ! R de…ned by g(x) = ¹_"f (^"x). It is immediately checked that g is 1-Lipschitz, so by the result above there exists a C¹ smooth, C0-Lipschitz function h such that jg(x) h(x)j 1 for all x, which implies that the function K(y) := "h(_"y) is C₀ -Lipschitz and satis…es jf(y) K(y)j " for all y 2 X.

We next establish the existence of a continuous and bounded selection of the Hahn-Banach extension operator y 2 Y ! G(y ) 2 2^X ; where

G(y ) = fx 2 X : x (y) = y (y) for all y 2 Y g:

Lemma 2. For every Banach space X and every closed subspace Y X there exist a continuous mapping H : Y ! X and a number M 1 such that

(1) H(y )(y) = y (y) for every y 2 Y , y 2 Y ; (2) kH(y )kX M ky kY for every y 2 Y .

Proof. This is a consequence of the Bartle-Graves selector theorem (see [DGZ, page 299]) which states: Let W and Z be Banach spaces and let T be a bounded linear mapping of W onto Z. Then there exists a continuous (nonlinear in general) mapping B of Z into W such that (T B) w = w for every w 2 W . Moreover, it follows from the proof of this result that there exists an M > 1 such that kB(w)k M kwk. If we apply this theorem with W = X , Z = Y , to the mapping T : X ! Y de…ned by T (x ) = x_jY, which is a continuous linear surjection with kT k = 1 (by the Hahn-Banach theorem), we obtain our continuous map H = B : Y ! X with the property that the the restriction of H(y ) to Y is y , for every y 2 Y , and such that kH(y )kX M ky kY .

Now we are in a situation to deduce an approximation result which is of independent interest and which, combined with some ideas of the Tietze proof, will yield our main results on smooth extension.

Theorem 1. Let X be a separable Banach space which admits a C¹- smooth norm, and Y X a closed subspace. Let f : Y ! R be a C¹-smooth function, and F a continuous extension of f to X. Let H : Y ! X be any extension operator as in Lemma 2. Then, for every " > 0, there exists a C¹-smooth map g : X ! R such that

(1) jF (x) g (x)j < " on X; and (2) kH(f⁰(y)) g⁰(y)kX < " on Y .

Furthermore, if the given C¹ function f is Lipschitz on Y and F is a Lipschitz extension of f to X with Lip(F ) = Lip(f ) (for instance F (x) = inf_y2Yff(y) + Lip(f)kx ykg), then the function g can be chosen to be Lipschitz on X and with the additional property that

(3) Lip(g) CLip(f );

where C > 1 is a constant only depending on X.

Proof. First note that by the Tietze Theorem, the continuous extension F always exists. We modify the proof of Theorem 4 in [AFGJL] employing Lemma 1. It will be convenient to use the following notation: given a point y_k2 Y; we de…ne Tk to be the natural H-extension of the …rst order Taylor Polynomial of f at y_k; namely, T_k(x) = f (y_k) + H(f⁰(y_k)) (x y_k). Note in particular that T_k2 C¹(X; R), with Tk⁰(x) = H(f⁰(y_k)) for every x 2 X, and T_k⁰(y) jY= f⁰(y_k) for all y 2 Y .

Now, using the separability of X; the closedness of Y X; and the continuity of F; we can construct a covering C = B^rj

1

j=1[ fB^skg¹k=1 of X, by open balls with centres x_j and y_k respectively, with the following properties:

(i). We have B_2rj XnY; and jF (x) F (x_j) j < "=2C0 on B_2rj, (ii). The collection fB^skgk X covers Y with centres yk 2 Y and radii s_k chosen using the smoothness of f on Y and the norm-norm conti- nuity of the extension operator H, so that kTk⁰(y) f⁰(y)k_Y < "=8C₀ and kTk⁰(y) H(f⁰(y))k_X < "=8C₀ on B_2sk\ Y:

It will be useful in the sequel, to employ an alternate notation for the open balls B_rj and B_sk: We let : N ! C be a bijection where for each i, (i) = B( 1(i); 2(i)). Let 'j 2 C¹(X; [0; 1]) with bounded derivative so that '_j = 1 on B( ₁(j); ₂(j)) and '_j = 0 outside of B( ₁(j); 2 ₂(j)).

By Lemma 1 applied to T_k(y) f (y) on B_2sk\ Y; we may choose C¹-smooth maps _k: X ! R so that on each B^2sk\ Y we have both

jTk(y) f (y) _k(y)j < 2 ^{k 2}"M_k¹; and k k⁰ (y)k_X < "=8, where M_k=Pk

i=1Mf_iand fM_i = sup_{x2Y \B}

2sik'⁰i(x)kX . Then we also have, for y 2 B^2sk\ Y using our estimate above,

T_k⁰(y) H(f⁰(y)) ⁰_k(y) _X T_k⁰(y) H f⁰(y) _X + _k⁰ (y) _X

< "=8C0+ "=8 "=4:

Set _i(x) = T_k(x) _k(x) if (i) = B_sk is a ball from the subcollec- tion fBslg¹_l=1 covering Y , and _i = F (x_j) if (i) = B_rj belongs to the subcollection fBrlg¹l=1 covering X n Y .

Next, we de…ne

h_i= '_iY

k<i

(1 '_k) ;

and

g (x) =X

i

h_i(x) _i(x)

Note that for each x, if n := n (x) := min fm : x 2 (m)g, then be- cause 1 'n(x) = 0 and (n) is open, it follows from the de…nition of the h_j that there is a neighbourhood N (n) of x so that for z 2 N, g (z) =P

j nhj(z) j(z), andP

jhj(z) =P

j nhj(z). Also, by a straight- forward calculation, again using the fact that 'n= 1 on (n), we have that P

jh_j(z) = 1 for z 2 (n), and so for all z 2 X:

Now, …x any x₀ 2 X; and let n0 = n (x₀) and a neighborhood N₀ of x₀ be as above. For each j n₀ de…ne the functions V_j : N₀ ! R and Wj : N0! R by

V_j(x) = 0 if ₁(j) =2 Y

jT^k(x) F (x) k(x)j if ¹(j) = yk

and

Wj(x) = 0 if 1(j) 2 Y

jF (xⁱ) F (x)j if ¹(j) = xi

Then for any x 2 N0 we have that jg (x) F (x)j X

j n0

hj(x) maxfV^j(x); Wj(x)g

X

j n0

h_j(x)"

2 < ":

Now de…ne the function so that when ₁(j) 2 Y; 1(j) = y _(j). Recall that B_2rj \ Y = ; for every j; and that 'j = 0 o¤ of B_2rj: Hence, if y 2 Y;

then in the sum g (y) only those indices j such that ₁(j) 2 Y are non-zero.

Recall also that P

jh_j(x) = 1 for all x 2 X (and henceP

jh⁰_j(x) = 0); and so for y 2 Y we have, H (f⁰(y)) = P

jh⁰_j(y) f (y) +P

jh_j(y) H (f⁰(y)) : And, g⁰(y) =P

jh⁰_j(y) T _(j)(y) _(j)(y) +P

jhj(y) T⁰_(j)(y) ⁰_(j)(y) : Finally, a straightforward calculation shows that h⁰_j(x) M _(j)for 1(j) 2 Y . With these observations in mind, we have,

g⁰(y) H f⁰(y) _X X

j n(y)

1(j)2Y

h⁰_j(y) T _(j)(y) f (y) _(j)(y)

+ h_j(y) T⁰_(j) (y) H(f⁰(y)) ⁰_(j)(y)

X

< X

j n(y)

1(j)2Y

h⁰_j(y) 2 ^{(j) 2}"M _(j)¹ + X

j n(y)

1(j)2Y

h_j(y)"

4

< X

j n(y)

1(j)2Y

M _(j) 2 ^{(j) 2}"M _(j)¹ + "

4 < ":

As H (f⁰(y)) jY= f⁰(y) ; we also have the estimate kg⁰(y) f⁰(y)kY < ":

Let us now consider the case when f is C¹ and Lipschitz on Y and F is any Lipschitz extension of f to X with Lip(F ) = Lip(f ). In this case we have to modify the de…nition of the functions i as follows.

We let _i(x) = T_k(x) _k(x) if (i) = B_sk is a ball from the subcol- lection fBslg¹l=1 covering Y where _k is chosen (by using Lemma 1) so that jTk(y) f (y) _k(y)j < 2 ^{i 2}"M_k¹, and k _k⁰ (y)k_X < "=8, where now the M_i are de…ned by M_i=Pi

j=1Mf_j and fM_j = sup_{x2B( 1(j);2 2(j))} '⁰_j(x)

X . We also let i(x) = F`(x) if (i) = Br`belongs to the subcollection fB^rjg¹j=1

covering X n Y , where the function F` is again chosen by using Lemma 1 so that jF`(x) F (x)j < 2 ^{i 2}"M_l ¹ on B_2rl and Lip(F_`) C₀Lip(F ) = C0Lip(f ). Note that, with these choices, we have

j i(x) F (x)j < 2 ^{i 2}"M_i ¹ on B( ₁(i); ₂(i)); and Lip( _i) maxfMLip(f) +"

8; C₀Lip(f )g;

where M is as in Lemma 2(2).

We can obviously assume f is not constant, so that Lip(f ) > 0, and if "

is chosen small enough the last inequality implies Lip( _i) C₁Lip(f );

where C₁ is a constant depending only on X.

Now de…ne the C¹ function g : X ! R by g(x) =X

i

i(x)hi(x):

As above, one can check that jg(x) F (x)j < " for all x 2 X, and also kg⁰(y) H(f⁰(y))kX < " for all y 2 Y ; that is g satis…es properties (1) and (2) of the statement. Let us see that g satis…es (3) as well. Noting that Lip(h_j) M_j, we can estimate, for every x; z 2 X,

g(x) g(z)

=X

j

j(x)hj(x) X

j

j(z)hj(z)

=X

j

( _j(x) F (x))(h_j(x) h_j(z)) +X

( _j(x) _j(z))h_j(z) X

j

"

2^j+2M_jLip(h_j)kx zk +X

j

Lip( _j)kx zkhj(z)

"

4 + C1Lip(f ) kx zk CLip(f )kx zk;

provided that " > 0 is chosen small enough (recall that we are assuming Lip(f ) > 0), and where C = 2C₁ > 1, a constant only depending on X.

This shows that Lip(g) C Lip(f ).

Theorem 2. Let X be a separable Banach space which admits a C¹- smooth norm. Let Y X be a closed subspace, and f : Y ! R a C¹- smooth and Lipschitz function. Then there is a C¹ and Lipschitz extension g : X ! R of f such that Lip(g) CLip(f ), where C is a constant depending only on X.

Proof. First note that if h is a bounded, Lipschitz function de…ned on Y , there always exists a bounded, Lipschitz extension of h to X, with the same Lipschitz constant, and bounded by the same constant (de…ned for instance by x 7! maxf khk1; minfkhk1; inf_y2Yfh(y) + Lip(h)kx ykg g g). For the purposes of the proof, we denote such an extension by h.

We are going to de…ne our function g by means of a series constructed by induction. By Theorem 1 there exists a C¹ function g1 : X ! R such that

jf g1j < 2 ¹" on X,

kf⁰(y) g₁⁰ (y)kY < 2 ¹"=C for y 2 Y (note in particular that this implies Lip(f g_jY) 2 ¹"=C), and

Lip (g1) C Lip(f )

Now, for n 2, suppose that we have chosen g1; :::; gn; real-valued and C¹-smooth on X such that for all x 2 X and y 2 Y;

f Xn i=1

gi

!

(x) < 2 ⁿ";

f⁰(y) Xn

i=1

g⁰_i(y)

Y

< 2 ⁿ"=C;

and

Lip(g_n) C Lip 0

@f (

n 1X

j=1

g_j)_jY 1 A :

It is clear that an application of Theorem 1 to the function f (g₁)_jY provides us with a function g2 which, together with g1, makes the above properties true for n = 2. Hence we can proceed to the general step of our inductive construction.

Consider the function l = f Pn

i=1g_i; which is C¹-smooth on Y:

By Theorem 1, we can …nd a C¹-smooth map g_n+1 on X such that we have,

(2.1) l (x) g_n+1(x) < 2 ^{n 1}" on X;

and for y 2 Y;

(2.2) l⁰(y) g_n+1⁰ (y) _Y < 2 ^{n 1}"=C;

and also,

(2.3) Lip (gn+1) C Lip (l) :

From (2:5) ; we have in particular, f (y) Pn+1

i=1 g_i(y) = jl (y) g_n+1(y)j <

2 ^{n 1}" on Y; and so f Pn+1

i=1 g_i (x) < 2 ^{n 1}" on X: This together with (2:6) and (2:7) completes the inductive step.

Now, from (2:6) we have Lip f (Pn+1

i=1 gi)_jY 2 ⁿ"=C, and so from (2:7) we obtain,

kg⁰n+1(x)k Lip(gn+1) C Lip 0

@f ( Xn j=1

gj)_jY 1

A C2 ⁿ"=C = 2 ⁿ":

Hence the series P

jg_j⁰(x) is absolutely and uniformly convergent on X.

Similarly, we have the estimate jgⁿ⁺¹(x)j 2 ⁿ⁺¹". Therefore the series g(x) =

X1 n=1

gn(x)

de…nes a C¹function on X, which coincides with f on Y because of the …rst inequality in the inductive assumptions. Finally, we have

Lip(g) Lip(g1) + X1 n=2

Lip(gn) C Lip(f ) + X1 n=2

2 ^{(n 1)}" 2C Lip(f );

provided that Lip(f ) > 0 (which we can always assume) and " is small enough.

Remark 1. With some more work in the proofs of the preceding the- orems one could show that the constant C can be taken to be any number C > M , where M is as in Lemma 2(2). Unfortunately the proof of the Bartle-Graves extension theorem does not give us any useful estimation about the size of M , and in general M is going to be quite large, so we cannot hope that any re…nement of the above proofs will yield a statement of Theorem 2 in which C can be chosen to be any number bigger than 1.

Theorem 3. Let X be a separable Banach space which admits a C¹- smooth norm. Let Y X be a closed subspace, and f : Y ! R a C¹-smooth function. Then there is a C¹ extension of f to X.

Proof. Since f is C¹on Y , there exists fB^jg := fB(y^j; rj)g¹j=1, a countable covering of Y by open balls in X such that f is Lipschitz on B_j\ Y for each j 2 N. Let U =S₁

j=1Bj, and W = X n Y . Consider the mapping h : X ! X de…ned by

h(x) = 1 1 + kxkx:

It is easily checked that h is a C¹di¤eomorphism from X onto its open unit ball intB_X (with inverse h ¹(y) = (1=(1 kyk)) y), that h has a bounded derivative, and that h preserves lines and in particular leaves the subspace Y invariant. By composing h with suitable dilations and translations we get C¹ di¤eomorphisms hj : X ! B^j such that hj is Lipschitz for each j.

And, by composing the restrictions to Y of these h_j with our function f , we get C¹ and Lipschitz functions f_j := f (h_j)_jY : Y ! R. According to the preceding result there exist C¹ (and Lipschitz) extensions G_j : X ! R of fj. Then the composition

g_j = G_j h_j¹ de…nes a C¹ extension of f_j

Bj \Y to B_j. Put g₀ 1:

Now let f'0g [ f'jg¹j=1 be a C¹ partition of unity subordinated to the open covering fW g [ fBjg¹j=1 of X (such partitions of unity always exist for separable spaces with C¹ norms, see [DGZ, Theorem VIII.3.2, page 351]).

De…ne

g(x) = X1 j=0

'j(x)gj(x):

Then it is clear that g is a C¹ extension of f to X.

Corollary 1. Let M be a separable Banach manifold modelled on a Banach space X which admits a C¹ norm, and let N be a closed C¹ sub- manifold of M . Then every C¹ function f : N ! R has a C¹ extension to M .

Proof. Let fV^jg¹j=1 be a covering of N by open sets in M so that there are C¹ di¤eomorphisms j : Vj ! X such that ^j(N \ V^j) = Y , where Y is a closed subspace of X.

The functions f _j¹ : Y ! R are C¹ and (by the preceding theorem) there are C¹ extensions Gj : X ! R, which in turn give, by composition, C¹ extensions gj := Gj j of f_{jVj \N} to Vj.

Then, if f g[f ^jg¹j=1is a C¹ partition of unity subordinated to the open covering fM n Ng [ fVjg¹j=1 of M (note that a separable Banach manifold modelled on a Banach space X admits C¹ partitions of unity if and only if X does), the function

g(x) =X

j

j(x)g_j(x) is a C¹ extension of f to M .

If Y is not required to be a closed subspace of X but is merely closed, results similar to Theorem 1 and Theorem 2 can be obtained. However, the di¤erentiability requirements on f must be strengthened. The proofs, which we omit, closely parallel those for Theorem 1 and Theorem 2, where the essential di¤erence is that f⁰(y) is extended to directions o¤ of Y , not by the Bartle-Graves generated H(f⁰(y)), but by explicit hypothesis.

Theorem 4. Let X be a separable Banach space which admits a C¹- smooth norm, Y X a closed subset, and U Y a neighbourhood of Y: Let

" > 0; and f : U ! R be a map which is C¹-smooth on Y as a function on X: Then there exists a C¹-smooth map g : X ! R such that,

(1) jf (y) g (y)j < " on Y;

(2) kf⁰(y) g⁰(y)kX < " on Y:

Theorem 5. Let X be a separable Banach space which admits a C¹- smooth norm. Let Y X be a closed subset, and U an open set containing Y: Let f : U ! R be a map C¹-smooth on Y as a function on X: Then there is a C¹ extension of f j^Y to X.

We have the following easy corollary.

Corollary 2. Let X be a separable Banach space which admits a C¹- smooth norm. Let U X be open and f : U ! R a C¹-smooth function.

Then for any open set V U with V U; there is a C¹ extension of f jV

to X.

Acknowledgement. The authors wish to thank Richard Aron, Gilles Godefroy, and Sophie Grivaux for many helpful discussions. We also thank P. Hajek and M. Johannis for drawing our attention to their recent work [HJ], which led to a signi…cant improvement of the …rst version of our paper.

References

[A] R. Aron, private communication.

[At] C.J. Atkin, Extension of smooth functions in in…nite dimensions I: unions of convex sets, Studia Math., 146 (3) (2001), 201-226.

[AB] R. Aron and P. Berner, A Hahn-Banach extension theorem for analytic maps, Bull. Soc. Math. France., 106 (1978), 3-24.

[AFM] D. Azagra, R. Fry, and A. Montesinos, Perturbed Smooth Lipschitz Extensions of Uniformly Continuous Functions on Banach Spaces, Proc. Amer. Math. Soc., 133(2005), 727-734.

[AFGJL] D. Azagra, R. Fry, J. Gómez Gil, J.A. Jaramillo, and M. Lovo, C¹-…ne approxi- mation of functions on Banach spaces with unconditional basis, Oxford Quarterly J. Math., 56 (2005), 13-20.

[BMP1] E. Bierstone, P. Milman, and W. Pawluka, Di¤ erentiable functions de…ned in closed sets. A problem of Whitney, Invent. Math., 151 (2003), 329-352.

[BMP2] E. Bierstone, P. Milman, and W. Pawluka, Higher order tangents and Fe¤ erman’s paper on Whitney’s extension problem, Ann. of Math., (2006), 361-370.

[BS1] Y. Brudnyi and P. Shvartsman, Generalizations of Whitney’s extension theorem, IMRN, 3 (1994), 1-11.

[BS2] Y. Brudnyi and P. Shvartsman, Whitney’s extension problem for multivariate C^1;!-functions, Trans. Amer. Math. Soc., 353 (2001), 2487-2512.

[DGZ] R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, vol. 64, Pitman Monographs and Surveys in Pure and Applied Mathe- matics, 1993.

[Fe1] C. Fe¤erman, C^m extension by linear operators, Ann. of Math. 166 (2007), 779-835.

[Fe2] C. Fe¤erman, Whitney’s extension problem for C^m;Ann. of Math. 164 (2006), 313-359.

[Fe3] C. Fe¤erman, A sharp form of Whitney’s extension theorem, Ann. of Math. 161 (2005), 509-577 .

[FHHMPZ] M. Fabian, P. Habala, P. Hájek, V.M. Santalucía, J. Pelant, and V. Zizler, Functional analysis and in…nite-dimensional geometry, CMS books in mathe- matics 8, Springer-Verlag, 2001.

[F] R. Fry, Approximation by functions with bounded derivative on Banach spaces, Bull. Australian Math. Soc. 69 (2004), 125-131.

[G] G. Glaeser, Etudes de quelques algebres tayloriennes, J. d’Analyse., 6 (1958), 1-124.

[HJ] P. Hájek and M. Johanis, Smooth approximations, preprint.

[LT] J. Lindenstrauss, L. Tzafriri, On the complemented subspaces problem, Israel J.

Math. 9 (1971), 263–269.

[M] N. Moulis, Approximation de fonctions di¤ érentiables sur certains espaces de Banach, Ann. Inst. Fourier, Grenoble 21, 4 (1971), 293-345.

[W1] H. Whitney, Analytic extensions of di¤ erential functions in closed sets, Trans.

Amer. Math. Soc. 36 (1934), 63-89.

[W2] H. Whitney, Di¤ erentiable functions de…ned in closed sets I, Trans. Amer. Math.

Soc. 36 (1934), 369-387.

[Wi] S. Willard, Topology, Addison-Wesley, 1970.

[Z] V. Zizler, Smooth extensions of norms and complementability of subspaces, Arch.

Math. 53 (1989), 585-589.

ICMAT (CSIC-UAM-UC3-UCM), Departamento de Análisis Matemático, Fac- ultad Ciencias Matemáticas, Universidad Complutense, 28040, Madrid, Spain

E-mail address : [email protected]

Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, B.C., Canada

E-mail address : [email protected]

Department of Mathematics and Statistics, University of Northern British Columbia, Prince Georege, B.C., Canada

E-mail address : [email protected]