Orthogonality in normed spaces

TL;DR

This paper investigates the properties of orthogonality and metric projections in normed spaces, addressing questions related to Fredholm stability and strictly singular operators, and presents new insights possibly overlooked in existing literature.

Contribution

It provides new results on orthogonality in normed spaces and explores properties of metric projections, filling gaps in the literature and posing open questions.

Findings

Analysis of orthogonality properties in normed spaces

Characterization of metric projections in Banach spaces

Identification of open problems in the theory of Fredholm stability

Abstract

Motivated by the questions in the theory of Fredholm stability in Banach space and Kato's strictly singular operators we answer several natural questions concerning ``orthogonality'' in normed spaces and the properties of metric projections. What the reader will see below might have benn known long ago, but we did not find it in the literature. Some open (for us) questions are formulated at the end of Sections 7 and 9.