A Survey on the Complemented Subspace Problem

TL;DR

This survey reviews the complemented subspace problem in Banach space theory, discussing key results, historical context, and open questions about which subspaces are complemented.

Contribution

It provides a comprehensive overview of the complemented subspace problem, including historical development, recent results, and open challenges in Banach space theory.

Findings

Summarizes key results on complemented subspaces

Highlights open problems in the field

Provides historical context and development of the theory

Abstract

The complemented subspace problem asks, in general, which closed subspaces $M$ of a Banach space $X$ are complemented; i.e. there exists a closed subspace $N$ of $X$ such that $X=M\oplus N$? This problem is in the heart of the theory of Banach spaces and plays a key role in the development of the Banach space theory. Our aim is to investigate some results on complemented subspaces, to give a history of the subject, and to present some open problems.