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Approximation by Cp -smooth, Lipschitz functions on Banach spaces
J. Math. Anal. Appl. 315 (2006) 599–605 www.elsevier.com/locate/jmaa
Approximation by C p -smooth, Lipschitz functions on Banach spaces ✩ R. Fry ∗ Department of Maths, St. F. X. University, Antigonish, NS, Canada Received 1 July 2004 Available online 24 August 2005 Submitted by P.G. Casazza
Abstract If X is a Banach space with an unconditional basis and admits a C p -smooth, Lipschitz bump function, and Y is a subset of X, then any uniformly continuous function on Y can be uniformly approximated by functions which are C p -smooth and Lipschitz on X. 2005 Elsevier Inc. All rights reserved. Keywords: Smooth approximation; Banach spaces
1. Introduction The problem of uniformly approximating continuous functions by smooth functions on Banach spaces has received much attention over the last several decades. More precisely, if f : X → R is a continuous map on a Banach space X which admits a C p -smooth bump function (a C p -smooth function with bounded, non-empty support), then the problem is to find, for each ε > 0, a C p -smooth function g : X → R with |f (x) − g(x)| < ε for x ∈ X. Such results are known for quite general X, for example, when X is weakly compactly generated (see, e.g., [7]). Results of this kind are usually achieved by employing smooth ✩
Research partly supported by NSERC (Canada).
* Present address: Department of Mathematics, Thompson Rivers University, Kamloops, BC, Canada.
E-mail address: [email protected]. 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.07.009
600
R. Fry / J. Math. Anal. Appl. 315 (2006) 599–605
partitions of unity. Often it is desirable that the approximating function g have some nice properties in addition to basic smoothness, and frequently one is unable to use partitions of unity to achieve this. For example, if X is superreflexive and f is Lipschitz, then it is shown in [4] (see also [8]) that when approximating on bounded sets, g can be chosen to have uniformly continuous derivative. Recently it was shown in [3] (see also [9]) that on separable spaces admitting a C p smooth, Lipschitz bump function, if f is bounded and uniformly continuous, then the approximating function g can be chosen Lipschitz in addition to being C p -smooth. It was asked in [3] and independently by J. Jaramillo, if the condition that f be bounded could be removed from this result of [3] (the condition that f be uniformly continuous is necessary here). The purpose of this note is to answer this question in the affirmative in the case when X admits an unconditional basis. Moreover, in the class of Banach spaces with unconditional basis, our main result characterizes X admitting C p -smooth, Lipschitz bump functions as those spaces where uniform approximation of uniformly continuous functions by C p -smooth, Lipschitz functions is possible. This characterization is analogous to the classical case, in the class of separable spaces, of X admitting C p -smooth bump functions [6]. The notation we employ is standard, with X, Y, etc., denoting Banach spaces. Smoothness in this note is meant in the Fréchet sense. As pointed out above, a C p -smooth bump function b on X is a C p -smooth, real-valued function on X with bounded, non-empty support, where support(b) = {x ∈ X: b(x) = 0}. If f : X → Y is Lipschitz with constant η, we will say that f is η-Lipschitz. Most additional notation is explained as it is introduced in the sequel. For any unexplained terms we refer the reader to [7,8].
2. Main results We first introduce some notation which will be used throughout the paper. Let {ej , ej∗ }∞ Schauder basis on X, and Pn : X → X the canonical proj =1 be an unconditional n jections given by Pn (x) = Pn ( ∞ j =1 xj ej ) = j =1 xj ej , and where we set P0 = 0. Let the unconditional basis constant be C 1. In particular, Pn C for all n. We put En = Pn (X), and E ∞ = n En , noting that dim En = n and E ∞ = X. The proof of our main theorem is a modification of some techniques found in [2,11], where C p -fine approximation on Banach spaces is considered. We include here the main parts of the construction, and shall refer the reader to [2] in the appropriate places to avoid excessive technical detail. Theorem 1. Let X be a Banach space with unconditional basis which admits a C p -smooth, Lipschitz bump function. Let Y ⊂ X be a subset and f : Y → R a uniformly continuous map. Then for each ε > 0 there exists a C p -smooth, Lipschitz function g : X → R such that for all y ∈ Y , f (y) − g(y) < ε. Moreover, if X admits a C p -smooth norm with unconditional constant C, and if f is Lipschitz with constant η, then we can choose g to have Lipschitz constant 12C(1 + C)η.
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Proof. The main idea of the proof is a modification of the proof of [2, Lemma 5] whereby we need only assume that f is Lipschitz and not also C 1 -smooth. We will need to use the following result, and refer the reader to [7, Proposition II.5.1] and [10] for a proof. Proposition 1. Let Z be a Banach space. The following assertions are equivalent. (1) Z admits a C p -smooth, Lipschitz bump function. (2) There exist numbers a, b > 0 and a Lipschitz function ψ : Z → [0, ∞) which is C p -smooth on Z \ {0}, homogeneous (that is ψ(tx) = |t|ψ(x) for all t ∈ R, x ∈ Z), and such that a · ψ b · . For such a function ψ , the set A = {z ∈ Z: ψ(z) 1} is what we call a C p -smooth, Lipschitz starlike body, and the Minkowski functional of this body, µA (z) = inf{t > 0: (1/t)z ∈ A}, is precisely the function ψ (see [1] and the references therein for further information on starlike bodies and their Minkowski functionals). We will denote the open ball of center x and radius r, with respect to the norm · of X, by B(x, r). If A is a bounded starlike body of X, we define the open A-pseudoball of center x and radius r as BA (x, r) := y ∈ X: µA (y − x) < r . According to Proposition 1 and the preceding remarks, because X has a C p -smooth, Lipschitz bump function, there is a bounded starlike body A ⊂ X (which we fix for the remainder of the proof) whose Minkowski functional µA = ψ is Lipschitz and C p -smooth 1 on X \ {0}, and there is a number M 1 such that M x µA (x) Mx for all x ∈ X, and µA (x) M for all x ∈ X \ {0}. Notice that in this case we have, r ⊆ BA (x, r) ⊆ B(x, Mr) (2.1) B x, M for every x ∈ X, r > 0. This fact will sometimes be used implicitly in what follows. For the proof, we shall first define a function f¯ : E ∞ → R, then a map Ψ : X → E ∞ , and finally our desired function g will be given by g = f¯ ◦ Ψ . First, since every uniformly continuous function can be uniformly approximated by Lipschitz functions (see [5] for instance), we may and do assume that f is Lipschitz with constant, say, η. Let ε > 0 and r ∈ (0, ε/3CMη). Let ϕ : R → [0, 1] be a C ∞ -smooth function such that ϕ(t) = 1 if |t| < 1/2, ϕ(t) = 0 if |t| > 1, ϕ ([0, ∞)) ⊆ [−3, 0], ϕ(−t) = ϕ(t). Let F be a Lipschitz extension of f from Y to X, with the same constant η (defined for instance by F (x) = inf{f (y) + ηx − y: y ∈ Y }). Let us define a function on X, C p -smooth on En , by
(an )n Fn (x) = F (x − y)ϕ an µA (y) dy, cn En
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R. Fry / J. Math. Anal. Appl. 315 (2006) 599–605
where
cn =
ϕ µA (y) dy,
En
and (keeping in mind (2.1)) we have chosen the constants an large enough that ε sup Fn (x) − F (x) < . 3 x∈En Note that for any x, x ∈ X, n Fn (x) − Fn (x ) (an ) cn
(2.2)
F (x − y) − F (x − y)ϕ an µA (y) dy
En
ηx − x
(an )n cn
ϕ an µA (y) dy = ηx − x ,
En
that is, Fn is η-Lipschitz. Consider the following function on X, C p -smooth on En , defined by
(bn )n fˆn (x) = Fn (x − y)ϕ bn µA (y) dy, cn En
where cn is as above, and (again taking note of (2.1)) we have chosen the constants bn large enough so that ε sup fˆn (x) − Fn (x) < 2−n , 6 x∈En and also,
sup fˆn (x) − Fn (x) < η2−n .
x∈En
We next define a sequence of functions f¯n : X → R as follows. Put f¯0 = f (0), and supposing that f¯0 , . . . , f¯n−1 have been defined, we set
f¯n (x) = fˆn (x) + f¯n−1 Pn−1 (x) − fˆn Pn−1 (x) . One can verify by induction that, (i) the restriction of f¯n to En is C p -smooth and f¯n extends f¯n−1 ; (ii) supx∈En |f¯n (x) − Fn (x)| < 3ε (1 − 21n ); (iii) supx∈En f¯n (x) − Fn (x) < 2η(1 − 21n ). We now define the map f¯ : E ∞ → R by f¯(x) = lim f¯n (x). n→∞
Note that for x ∈ E ∞ = ¯ fnx (x).
n En ,
there exists an nx so that f¯(x) = limn→∞ f¯n (x) =
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603
With the above notation in mind, one can verify using (i)–(iii) above that f¯ has the following properties: (i) the restriction of f¯ to every subspace En is C p -smooth; (ii) for each x ∈ E ∞ , |f¯(x) − Fnx (x)| 3ε ; (iii) for each x ∈ E ∞ , f¯ (x) − Fn x (x) 2η.
∈ X and define the maps
µA (x − Pn−1 (x)) , χn (x) = 1 − ϕ r
Next let x =
and Ψ (x) =
n xn en
χn (x)xn en .
n
For any x0 , because Pn (x0 ) → x0 and the Pn are uniformly bounded, there exist a neighbourhood N0 of x0 and an n0 = nx0 so that χn (x) = 0 for all x ∈ N0 and n n0 and so Ψ (N0 ) ⊂ En0 . Thus, Ψ : X → E ∞ is a C p -smooth map whose range is locally contained in the finite-dimensional subspaces En . Using the fact that {en } is unconditional with constant C, one can show that (see [2, Fact 7]) x − Ψ (x) < CMr. (2.3) We next consider the derivative of Ψ . A straightforward calculation, using the facts: |ϕ (t)| 3, µA (x) M and (I − Pn−1 ) (x) 1 + C for all x, t, verifies that χn (x) 3M(1 + C)r −1 . Also, since (χn (x)xn ) = χn (x)xn + χn (x)en∗ , it follows that χn (x)(·)xn en + χn (x)en en∗ (·). (2.4) Ψ (x)(·) = n
n
Now, using (2.4), the estimate for χn above, and again using the fact that {en } is unconditional with constant C, it is shown in [2, Fact 7] that for all x ∈ X, Ψ (x) 4M 2 C(1 + C). Finally, we define g(x) = f¯(Ψ (x)). Note that g is C p -smooth on X, being the composition of C p -smooth maps. Now, for x ∈ E ∞ , by (ii) , choice of r, using (2.2), (2.3), and finally using that Fn is Lipschitz with constant η, we have for n > nΨ (x) ,
F (x) − g(x) Fn (x) − F (x) + Fn (x) − Fn Ψ (x) + f¯ Ψ (x) − Fn Ψ (x) ε/3 + ηCMr + ε/3 < ε. Because E ∞ is dense in X, and both F and g are continuous, the above estimate gives that for all x ∈ X, F (x) − g(x) ε. In particular, since F |Y = f , we have for y ∈ Y that |f (y) − g(y)| ε. Lastly we have, using property (iii) of f¯ and our estimates above, that for all x ∈ X,
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g (x) f¯ Ψ (x) Ψ (x)
Fn Ψ (x) Ψ (x) + 2η 4M 2 C(1 + C) (η + 2η)4M 2 C(1 + C) = 12M 2 C(1 + C)η. This establishes the first statement of the theorem, with the Lipschitz constant of g given by 12M 2 C(1 + C)η. For the second statement, if X admits a C p -smooth norm, then we may use this norm in place of the C p -smooth, Lipschitz function µA , and hence the proof may be carried out with the choice M = 1 from which the second statement follows. 2 Corollary 1. Let X have unconditional basis. Then the following are equivalent. (1) X admits a C p -smooth, Lipschitz bump function. (2) For every subset Y ⊂ X, uniformly continuous map f : Y → R, and ε > 0, there exists a C p -smooth, Lipschitz map g : X → R with |f − g| < ε on Y . Proof. That (1) ⇒ (2) is Theorem 1. Conversely, choose Y = X, f = · , and a C p smooth, Lipschitz map g : X → R with |f − g| < 1 on X. Let ξ : R → R be C ∞ -smooth and Lipschitz with, ξ(t) = 1 if t 1 and ξ(t) = 0 if t 2. Then b = ξ ◦ g is a C p -smooth, Lipschitz map with b(0) = 1 and b(x) = 0 when x 3. 2 Remarks. (i) For the second statement of Theorem 1, an examination of the proof shows that if Z is any Banach space and f : X → Z is η-Lipschitz, then we can choose g to be C p -smooth and 12C(1 + C)η-Lipschitz with g : X → Z. (ii) The Lipschitz constant for g given in the second statement of Theorem 1 is not the best possible (see open problem 2 below). Open problems. 1. For X which admit C p -smooth, Lipschitz bump functions, is Theorem 1 true if we assume that X admits only a Schauder basis, or even merely that X is separable? As noted in the introduction, positive results are known when X is separable under the additional hypothesis that f be bounded [3]. We remark here that for general nonseparable X, even uniform approximation by non-Lipschitz, C p -smooth maps is a long standing open question. 2. Under the hypothesis of Theorem 1 when p = 1, if f is η-Lipschitz and η > η, can we choose g to be η -Lipschitz? (suggested to the author by D. Azagra).
Acknowledgments The author thanks D. Azagra and J. Jaramillo for bringing to his attention the problems addressed in this note during the RSME–AMS 2003 conference in Sevilla.
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References [1] D. Azagra, T. Dobrowolski, On the topological classification of starlike bodies in Banach spaces, Topology Appl. 132 (2003) 221–234. [2] D. Azagra, R. Fry, J.G. Gill, J.A. Jaramillo, M. Lovo, C 1 -fine approximation of functions on Banach spaces with unconditional bases, Quart. J. Math. Oxford Ser. 56 (2005) 13–20. [3] D. Azagra, R. Fry, A. Montesinos, Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces, Proc. Amer. Math. Soc. 133 (2005) 727–734. [4] M. Cepedello Boiso, Approximation of Lipschitz functions by ∆-convex functions in Banach spaces, Israel J. Math. 106 (1998) 269–284. [5] Y. Benyamini, J. Lindenstrauss, Geometrical Nonlinear Functional Analysis, vol. I, Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000. [6] R. Bonic, J. Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966) 877–898. [7] R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math., vol. 64, Longman, Harlow, 1993. [8] M. Fabian, P. Habala, P. Hájek, V.M. Santalucía, J. Pelant, V. Zizler, Functional Analysis and InfiniteDimensional Geometry, CMS Books in Math., vol. 8, Springer-Verlag, Berlin, 2001. [9] R. Fry, Approximation by functions with bounded derivative on Banach spaces, Bull. Austral. Math. Soc. 69 (2004) 125–131. [10] M. Leduc, Densité de certains familles d’hyperplans tangents, C. R. Acad. Sci. Paris Sér. A 270 (1970) 326–328. [11] N. Moulis, Approximation de fonctions différentiables sur certains espaces de Banach, Ann. Inst. Fourier (Grenoble) 21 (1971) 293–345.
Approximation by C p -smooth, Lipschitz functions on Banach spaces ✩ R. Fry ∗ Department of Maths, St. F. X. University, Antigonish, NS, Canada Received 1 July 2004 Available online 24 August 2005 Submitted by P.G. Casazza
Abstract If X is a Banach space with an unconditional basis and admits a C p -smooth, Lipschitz bump function, and Y is a subset of X, then any uniformly continuous function on Y can be uniformly approximated by functions which are C p -smooth and Lipschitz on X. 2005 Elsevier Inc. All rights reserved. Keywords: Smooth approximation; Banach spaces
1. Introduction The problem of uniformly approximating continuous functions by smooth functions on Banach spaces has received much attention over the last several decades. More precisely, if f : X → R is a continuous map on a Banach space X which admits a C p -smooth bump function (a C p -smooth function with bounded, non-empty support), then the problem is to find, for each ε > 0, a C p -smooth function g : X → R with |f (x) − g(x)| < ε for x ∈ X. Such results are known for quite general X, for example, when X is weakly compactly generated (see, e.g., [7]). Results of this kind are usually achieved by employing smooth ✩
Research partly supported by NSERC (Canada).
* Present address: Department of Mathematics, Thompson Rivers University, Kamloops, BC, Canada.
E-mail address: [email protected]. 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.07.009
600
R. Fry / J. Math. Anal. Appl. 315 (2006) 599–605
partitions of unity. Often it is desirable that the approximating function g have some nice properties in addition to basic smoothness, and frequently one is unable to use partitions of unity to achieve this. For example, if X is superreflexive and f is Lipschitz, then it is shown in [4] (see also [8]) that when approximating on bounded sets, g can be chosen to have uniformly continuous derivative. Recently it was shown in [3] (see also [9]) that on separable spaces admitting a C p smooth, Lipschitz bump function, if f is bounded and uniformly continuous, then the approximating function g can be chosen Lipschitz in addition to being C p -smooth. It was asked in [3] and independently by J. Jaramillo, if the condition that f be bounded could be removed from this result of [3] (the condition that f be uniformly continuous is necessary here). The purpose of this note is to answer this question in the affirmative in the case when X admits an unconditional basis. Moreover, in the class of Banach spaces with unconditional basis, our main result characterizes X admitting C p -smooth, Lipschitz bump functions as those spaces where uniform approximation of uniformly continuous functions by C p -smooth, Lipschitz functions is possible. This characterization is analogous to the classical case, in the class of separable spaces, of X admitting C p -smooth bump functions [6]. The notation we employ is standard, with X, Y, etc., denoting Banach spaces. Smoothness in this note is meant in the Fréchet sense. As pointed out above, a C p -smooth bump function b on X is a C p -smooth, real-valued function on X with bounded, non-empty support, where support(b) = {x ∈ X: b(x) = 0}. If f : X → Y is Lipschitz with constant η, we will say that f is η-Lipschitz. Most additional notation is explained as it is introduced in the sequel. For any unexplained terms we refer the reader to [7,8].
2. Main results We first introduce some notation which will be used throughout the paper. Let {ej , ej∗ }∞ Schauder basis on X, and Pn : X → X the canonical proj =1 be an unconditional n jections given by Pn (x) = Pn ( ∞ j =1 xj ej ) = j =1 xj ej , and where we set P0 = 0. Let the unconditional basis constant be C 1. In particular, Pn C for all n. We put En = Pn (X), and E ∞ = n En , noting that dim En = n and E ∞ = X. The proof of our main theorem is a modification of some techniques found in [2,11], where C p -fine approximation on Banach spaces is considered. We include here the main parts of the construction, and shall refer the reader to [2] in the appropriate places to avoid excessive technical detail. Theorem 1. Let X be a Banach space with unconditional basis which admits a C p -smooth, Lipschitz bump function. Let Y ⊂ X be a subset and f : Y → R a uniformly continuous map. Then for each ε > 0 there exists a C p -smooth, Lipschitz function g : X → R such that for all y ∈ Y , f (y) − g(y) < ε. Moreover, if X admits a C p -smooth norm with unconditional constant C, and if f is Lipschitz with constant η, then we can choose g to have Lipschitz constant 12C(1 + C)η.
R. Fry / J. Math. Anal. Appl. 315 (2006) 599–605
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Proof. The main idea of the proof is a modification of the proof of [2, Lemma 5] whereby we need only assume that f is Lipschitz and not also C 1 -smooth. We will need to use the following result, and refer the reader to [7, Proposition II.5.1] and [10] for a proof. Proposition 1. Let Z be a Banach space. The following assertions are equivalent. (1) Z admits a C p -smooth, Lipschitz bump function. (2) There exist numbers a, b > 0 and a Lipschitz function ψ : Z → [0, ∞) which is C p -smooth on Z \ {0}, homogeneous (that is ψ(tx) = |t|ψ(x) for all t ∈ R, x ∈ Z), and such that a · ψ b · . For such a function ψ , the set A = {z ∈ Z: ψ(z) 1} is what we call a C p -smooth, Lipschitz starlike body, and the Minkowski functional of this body, µA (z) = inf{t > 0: (1/t)z ∈ A}, is precisely the function ψ (see [1] and the references therein for further information on starlike bodies and their Minkowski functionals). We will denote the open ball of center x and radius r, with respect to the norm · of X, by B(x, r). If A is a bounded starlike body of X, we define the open A-pseudoball of center x and radius r as BA (x, r) := y ∈ X: µA (y − x) < r . According to Proposition 1 and the preceding remarks, because X has a C p -smooth, Lipschitz bump function, there is a bounded starlike body A ⊂ X (which we fix for the remainder of the proof) whose Minkowski functional µA = ψ is Lipschitz and C p -smooth 1 on X \ {0}, and there is a number M 1 such that M x µA (x) Mx for all x ∈ X, and µA (x) M for all x ∈ X \ {0}. Notice that in this case we have, r ⊆ BA (x, r) ⊆ B(x, Mr) (2.1) B x, M for every x ∈ X, r > 0. This fact will sometimes be used implicitly in what follows. For the proof, we shall first define a function f¯ : E ∞ → R, then a map Ψ : X → E ∞ , and finally our desired function g will be given by g = f¯ ◦ Ψ . First, since every uniformly continuous function can be uniformly approximated by Lipschitz functions (see [5] for instance), we may and do assume that f is Lipschitz with constant, say, η. Let ε > 0 and r ∈ (0, ε/3CMη). Let ϕ : R → [0, 1] be a C ∞ -smooth function such that ϕ(t) = 1 if |t| < 1/2, ϕ(t) = 0 if |t| > 1, ϕ ([0, ∞)) ⊆ [−3, 0], ϕ(−t) = ϕ(t). Let F be a Lipschitz extension of f from Y to X, with the same constant η (defined for instance by F (x) = inf{f (y) + ηx − y: y ∈ Y }). Let us define a function on X, C p -smooth on En , by
(an )n Fn (x) = F (x − y)ϕ an µA (y) dy, cn En
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R. Fry / J. Math. Anal. Appl. 315 (2006) 599–605
where
cn =
ϕ µA (y) dy,
En
and (keeping in mind (2.1)) we have chosen the constants an large enough that ε sup Fn (x) − F (x) < . 3 x∈En Note that for any x, x ∈ X, n Fn (x) − Fn (x ) (an ) cn
(2.2)
F (x − y) − F (x − y)ϕ an µA (y) dy
En
ηx − x
(an )n cn
ϕ an µA (y) dy = ηx − x ,
En
that is, Fn is η-Lipschitz. Consider the following function on X, C p -smooth on En , defined by
(bn )n fˆn (x) = Fn (x − y)ϕ bn µA (y) dy, cn En
where cn is as above, and (again taking note of (2.1)) we have chosen the constants bn large enough so that ε sup fˆn (x) − Fn (x) < 2−n , 6 x∈En and also,
sup fˆn (x) − Fn (x) < η2−n .
x∈En
We next define a sequence of functions f¯n : X → R as follows. Put f¯0 = f (0), and supposing that f¯0 , . . . , f¯n−1 have been defined, we set
f¯n (x) = fˆn (x) + f¯n−1 Pn−1 (x) − fˆn Pn−1 (x) . One can verify by induction that, (i) the restriction of f¯n to En is C p -smooth and f¯n extends f¯n−1 ; (ii) supx∈En |f¯n (x) − Fn (x)| < 3ε (1 − 21n ); (iii) supx∈En f¯n (x) − Fn (x) < 2η(1 − 21n ). We now define the map f¯ : E ∞ → R by f¯(x) = lim f¯n (x). n→∞
Note that for x ∈ E ∞ = ¯ fnx (x).
n En ,
there exists an nx so that f¯(x) = limn→∞ f¯n (x) =
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603
With the above notation in mind, one can verify using (i)–(iii) above that f¯ has the following properties: (i) the restriction of f¯ to every subspace En is C p -smooth; (ii) for each x ∈ E ∞ , |f¯(x) − Fnx (x)| 3ε ; (iii) for each x ∈ E ∞ , f¯ (x) − Fn x (x) 2η.
∈ X and define the maps
µA (x − Pn−1 (x)) , χn (x) = 1 − ϕ r
Next let x =
and Ψ (x) =
n xn en
χn (x)xn en .
n
For any x0 , because Pn (x0 ) → x0 and the Pn are uniformly bounded, there exist a neighbourhood N0 of x0 and an n0 = nx0 so that χn (x) = 0 for all x ∈ N0 and n n0 and so Ψ (N0 ) ⊂ En0 . Thus, Ψ : X → E ∞ is a C p -smooth map whose range is locally contained in the finite-dimensional subspaces En . Using the fact that {en } is unconditional with constant C, one can show that (see [2, Fact 7]) x − Ψ (x) < CMr. (2.3) We next consider the derivative of Ψ . A straightforward calculation, using the facts: |ϕ (t)| 3, µA (x) M and (I − Pn−1 ) (x) 1 + C for all x, t, verifies that χn (x) 3M(1 + C)r −1 . Also, since (χn (x)xn ) = χn (x)xn + χn (x)en∗ , it follows that χn (x)(·)xn en + χn (x)en en∗ (·). (2.4) Ψ (x)(·) = n
n
Now, using (2.4), the estimate for χn above, and again using the fact that {en } is unconditional with constant C, it is shown in [2, Fact 7] that for all x ∈ X, Ψ (x) 4M 2 C(1 + C). Finally, we define g(x) = f¯(Ψ (x)). Note that g is C p -smooth on X, being the composition of C p -smooth maps. Now, for x ∈ E ∞ , by (ii) , choice of r, using (2.2), (2.3), and finally using that Fn is Lipschitz with constant η, we have for n > nΨ (x) ,
F (x) − g(x) Fn (x) − F (x) + Fn (x) − Fn Ψ (x) + f¯ Ψ (x) − Fn Ψ (x) ε/3 + ηCMr + ε/3 < ε. Because E ∞ is dense in X, and both F and g are continuous, the above estimate gives that for all x ∈ X, F (x) − g(x) ε. In particular, since F |Y = f , we have for y ∈ Y that |f (y) − g(y)| ε. Lastly we have, using property (iii) of f¯ and our estimates above, that for all x ∈ X,
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R. Fry / J. Math. Anal. Appl. 315 (2006) 599–605
g (x) f¯ Ψ (x) Ψ (x)
Fn Ψ (x) Ψ (x) + 2η 4M 2 C(1 + C) (η + 2η)4M 2 C(1 + C) = 12M 2 C(1 + C)η. This establishes the first statement of the theorem, with the Lipschitz constant of g given by 12M 2 C(1 + C)η. For the second statement, if X admits a C p -smooth norm, then we may use this norm in place of the C p -smooth, Lipschitz function µA , and hence the proof may be carried out with the choice M = 1 from which the second statement follows. 2 Corollary 1. Let X have unconditional basis. Then the following are equivalent. (1) X admits a C p -smooth, Lipschitz bump function. (2) For every subset Y ⊂ X, uniformly continuous map f : Y → R, and ε > 0, there exists a C p -smooth, Lipschitz map g : X → R with |f − g| < ε on Y . Proof. That (1) ⇒ (2) is Theorem 1. Conversely, choose Y = X, f = · , and a C p smooth, Lipschitz map g : X → R with |f − g| < 1 on X. Let ξ : R → R be C ∞ -smooth and Lipschitz with, ξ(t) = 1 if t 1 and ξ(t) = 0 if t 2. Then b = ξ ◦ g is a C p -smooth, Lipschitz map with b(0) = 1 and b(x) = 0 when x 3. 2 Remarks. (i) For the second statement of Theorem 1, an examination of the proof shows that if Z is any Banach space and f : X → Z is η-Lipschitz, then we can choose g to be C p -smooth and 12C(1 + C)η-Lipschitz with g : X → Z. (ii) The Lipschitz constant for g given in the second statement of Theorem 1 is not the best possible (see open problem 2 below). Open problems. 1. For X which admit C p -smooth, Lipschitz bump functions, is Theorem 1 true if we assume that X admits only a Schauder basis, or even merely that X is separable? As noted in the introduction, positive results are known when X is separable under the additional hypothesis that f be bounded [3]. We remark here that for general nonseparable X, even uniform approximation by non-Lipschitz, C p -smooth maps is a long standing open question. 2. Under the hypothesis of Theorem 1 when p = 1, if f is η-Lipschitz and η > η, can we choose g to be η -Lipschitz? (suggested to the author by D. Azagra).
Acknowledgments The author thanks D. Azagra and J. Jaramillo for bringing to his attention the problems addressed in this note during the RSME–AMS 2003 conference in Sevilla.
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