Uniformly Gâteaux Smooth Approximations On

  • UNIFORMLY GÂTEAUX
  • Published 2007

Abstract

Every Lipschitz mapping from c0(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C∞-Fréchet smooth. The main result of this note is a construction of uniform approximations of Lipschitz mappings from c0(Γ) into a Banach space Y , by means of Lipschitz mappings that are also uniformly Gâteaux (UG) smooth and C∞-Fréchet differentiable. We also construct an equivalent UG and C∞-Fréchet smooth renorming of c0(Γ). Finally, we construct an example of a convex, even, Lipschitz and UG-smooth separating function, such that the Minkowski functional of its sub-level set is not UG-smooth. The first two results answer problems posed in [FMZ]. An example of the implicit function method violating UG smoothness was lacking in the literature. Its existence is not surprising to the specialists, as the known constructions of UG renormings always take a detour around this otherwise standard method of obtaining smooth renormings. Let us recall that all Banach spaces admitting a UG-smooth bump function are in particular weak Asplund spaces by a fundamental result of Preiss, Phelps and Namioka in [PPN] (see also [F] or [BL]). However, the additional uniformity of the derivatives leads to a considerably stronger theory, with several characterisations of Banach spaces admitting a UG bump function (or a renorming). In particular, a Banach space admits an equivalent UG renorming if and only if its dual ball is a uniform Eberlein compact [FGZ]. For basic properties of UG smoothness we refer to [DGZ] and [F-Z]. To shed some light on the significance of our results, let us briefly summarise some of the more recent results concerning UG smoothness, defined below (for simplicity we assume that the domain is a whole Banach space X). It is shown in [LV] that a continuous UG-smooth real function on a Banach space X is locally Lipschitz. Moreover, if the function is uniformly continuous (or bounded), then it is globally Lipschitz. Thus some uniformity (Lipschitz) condition is in some sense also necessary for a mapping to be UG approximable. Tang [T] has shown that the existence of a UG bump function on a Banach space implies the existence of an equivalent UG renorming (analogous statement for Gâteaux smooth bumps is false [H]), and used this fact to show that every convex function on such a space is uniformly approximable by convex and UG-smooth functions on bounded sets. The more general problem of approximating all Lipschitz functions seems to be still open. One of the difficulties is that the standard approach to constructing smooth approximations by using smooth partitions of unity appears to be failing (loss of uniformity). In this regard let us mention that in the stronger uniformly Fréchet case it was shown by John, Toruńczyk and Zizler [JTZ] that the UF-smooth partitions of unity always exist provided the space has a UF bump function. However, the existence of UF approximations of Lipschitz functions seems to be open. In the separable setting, Fabian and Zizler [FZ] were able to combine the best Fréchet smoothness of the space in question together with the UG condition (recall that every separable Banach space has a UG renorming). Namely, if a separable Banach space admits a C-Fréchet smooth norm, than it admits also a norm which is simultaneously C-Fréchet smooth and UG. The techniques used in their paper are strongly separable in nature, which leads to the natural question of what happens in the general case. This is the source of the questions asked in [FMZ], resolved in our note. Let us now proceed with the preliminaries to our results. For an arbitrary set A, we denote its cardinality by |A|. For n ∈ N, λn denotes the Lebesgue measure on R. B(x, r) and U(x, r) are closed and open ball centred at x with diameter r. For F ⊂ Γ we denote the associated projection by PF , i.e. PF x = ∑ γ∈F e ∗ γ(x)eγ where x ∈ c0(Γ). By c00(Γ) we denote the linear subspace of c0(Γ) consisting of finitely supported vectors. The canonical supremum norm on c0(Γ) will be denoted by ‖·‖. We will say that a mapping f defined on a vector space X is even, if f(αx) = f(x) for all x ∈ X and all scalars α such that |α| = 1. Let X and Y be normed linear spaces, Ω ⊂ X be open and f : Ω → Y . We will denote the directional derivative of f at x ∈ Ω in the direction h ∈ X by Dhf(x) = limt→0 1t (f(x + th) − f(x)). If f is Gâteaux differentiable for all Date: November 2007. 2000 Mathematics Subject Classification. 46B03, 46B20, 46B26.

Cite this paper

@inproceedings{GTEAUX2007UniformlyGS, title={Uniformly G{\^a}teaux Smooth Approximations On}, author={UNIFORMLY G{\^A}TEAUX}, year={2007} }