Approximate smoothness in normed linear spaces

TL;DR

This paper introduces the concept of approximate smoothness in normed linear spaces, characterizes it, and explores its implications for Birkhoff-James orthogonality and right-additivity.

Contribution

It is the first to define and analyze approximate smoothness, linking it to classical smoothness and orthogonality properties in normed spaces.

Findings

Approximate smoothness is characterized in various normed spaces.

Connections between smoothness and approximate smoothness are established.

Approximate smoothness implies right-additivity of Birkhoff-James orthogonality.

Abstract

We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.