Linear relations and their singular chains

TL;DR

This paper explores the structure of singular chain spaces for linear relations in both finite and infinite-dimensional spaces, providing new characterizations and insights into their decomposition.

Contribution

It extends the understanding of singular chains and chain spaces to infinite-dimensional linear spaces, introducing identities involving root spaces.

Findings

Characterization of singular chain spaces via root spaces

Extension of finite-dimensional concepts to infinite-dimensional spaces

Insights into proper eigenvalues of linear relations

Abstract

Singular chain spaces for linear relations in linear spaces play a fundamental role in the decomposition of linear relations in finite-dimensional spaces. In this paper singular chains and singular chain spaces are discussed in detail for not necessarily finite-dimensional linear spaces. This leads to an identity characterizing a singular chain space in terms of root spaces. The so-called proper eigenvalues of a linear relation play an important role in the finite-dimensional case.