Smooth Functions and Partitions of Unity on Certain Banach Spaces

In an earlier paper [4], the author sketched a method, based on the use of “Talagrand operators”, for defining infinitely differentiable equivalent norms on the spaces C0(L) for certain locally compact, scattered spaces L. A special case of this result was that a C renorming exists on C0(L) for every countable locally compact L. Recently, Hájek [3] extended this result by showing that a real normed space X admits a C renorming whenever there is a countable subset of the unit ball of X∗ on which every element of X attains its norm, that is to say, a countable boundary. This suggested to the author that the locally compact topology on L was perhaps not essential in [4], and in the first part of the present paper we shall develop the methods of that work in a way that does not require such a topology. We obtain infinitely differentiable norms on certain (typically non-separable) Banach spaces X as well as on some certain injective tensor products X ⊗ǫ E. In the second part of the paper we present a lemma about partitions of unity. It is an open problem whether a non-separable Banach space with a C norm (or, more generally, a C “bump function”) admits C partitions of unity, though many partial results in this direction are known. Our lemma enables us to show that the answer is yes for classes of Banach spaces that admit projectional resolutions of the identity. In particular, we show that the space C0[0,Ω) admits C ∞ partitions of unity for every ordinal Ω. Results from this paper are used in [5] to give examples of Banach spaces admitting infinitely differentiable bump functions and partitions of unity but no smooth norms. Our notation and terminology are mostly standard and, whenever possible, we have followed the conventions of [2]. Although that work contains everything that the reader will need in order to understand the present paper, we recall for convenience a few facts and definitions. It should be noted that we are concerned only with real, as opposed to complex, Banach spaces. When we refer to a function on a Banach space as being of class C, where k is a positive integer, it is the standard (Fréchet) notion of smoothness that we are employing. Making a mild abuse of language, we shall say that a norm ‖ · ‖ on a Banach space X is of class C if the function x 7→ ‖x‖ is of that class on the set X \ {0}. (Of course, no norm is differentiable at 0.) A bump function on a Banach space X is a function φ : X → R with bounded, non-empty support. On finite-dimensional spaces, C bump functions are plentiful (and fundamental to the theory of distributions). The existence of a C bump function on an infinite-dimensional Banach space X is already a strong condition.