Smooth approximations of norms in separable Banach spaces

TL;DR

This paper proves that in separable Banach spaces with a smooth norm, any equivalent norm can be uniformly approximated on bounded sets by smooth norms, enhancing the understanding of norm smoothness and approximation.

Contribution

It establishes that all equivalent norms on such Banach spaces can be approximated uniformly on bounded sets by C^k-smooth norms, extending the class of norms known to be approximable.

Findings

Any equivalent norm can be approximated uniformly on bounded sets by C^k-smooth norms.

The result applies to Banach spaces with a k-times continuously Fréchet differentiable norm.

The approximation preserves the smoothness properties of the original norm.

Abstract

Let X be a separable real Banach space having a k-times continuously Fr\'{e}chet differentiable (i.e. C^k-smooth) norm where k=1,...,\infty. We show that any equivalent norm on X can be approximated uniformly on bounded sets by C^k-smooth norms.