Smooth norms and approximation in Banach spaces of the type C(K)

TL;DR

This paper investigates the smooth approximation properties of Banach spaces of continuous functions on compact spaces, establishing conditions under which functions can be approximated smoothly and norms can be refined for better smoothness.

Contribution

It proves new theorems linking the existence of smooth functions and norms on C(K) spaces, advancing understanding of their geometric and functional structure.

Findings

Continuous functions can be uniformly approximated by smooth functions under certain conditions.

Existence of a smooth norm is guaranteed if the dual norm is locally uniformly convex.

Theorems connect smooth function existence with geometric properties of Banach spaces.

Abstract

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class C^m. (ii) If C(K) admits an equivalent norm with locally uniformly convex dual norm, then C(K) admits an equivalent norm which is of class C^infty (except at 0).