# Some remarks on smooth renormings of Banach spaces

## TL;DR

This paper demonstrates that in separable Banach spaces with a Schauder basis and a smooth norm, any equivalent norm can be uniformly approximated by smoother norms with arbitrarily fast convergence depending on the basis tail.

## Contribution

It provides a method for approximating equivalent norms with smooth norms in Banach spaces, solving a problem posed in recent mathematical literature.

## Key findings

- Approximation of norms can be made arbitrarily fast.
- The approximation depends only on the tail of the Schauder basis.
- The result applies to all separable Banach spaces with a smooth norm.

## Abstract

We prove that in every separable Banach space $X$ with a Schauder basis and a $C^k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C^k$-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler.