A Geometry Characteristic for Banach Space with $c^1$-Norm

TL;DR

This paper establishes a geometric characterization of Banach spaces with a $c^1$-norm by linking the differentiability of the norm to the smooth manifold structure of the unit sphere.

Contribution

It proves that a Banach space's norm is $c^1$ if and only if its unit sphere forms a $c^1$-submanifold of codimension one, revealing a deep geometric connection.

Findings

Norm of $E$ is $c^1$ iff $S(E)$ is a $c^1$-submanifold

The proof links differentiability of the norm to the differential structure of the sphere

Provides a geometric criterion for $c^1$-norms in Banach spaces

Abstract

Let $E$ be a Banach space with the $c^1$-norm $\|\cdot\|$ in $ E \backslash \{0\}$ and $S(E)=\{e\in E: \|e\|=1\}.$ In this paper, a geometry characteristic for $E$ is presented by using a geometrical construct of $S(E).$ That is, the following theorem holds : the norm of $E$ is of $c^1$ in $ E \backslash \{0\}$ if and only if $S(E)$ is a $c^1$-submanifold of $E,$ with ${\rm codim}S(E)=1.$ The theorem is very clear, however, its proof is non-trivial, which shows an intrinsic connection between the continuous differentiability of the norm $\|\cdot\|$ in $ E \backslash \{0\}$ and differential structure of $S(E).$