Lineability and spaceability: a new approach

TL;DR

This paper introduces a novel approach to lineability and spaceability, exploring the existence of linear structures within complex subsets of vector spaces across various mathematical contexts.

Contribution

It presents a new methodology for investigating lineability and spaceability, expanding the scope beyond traditional restrictive approaches.

Findings

Positive results on lineability/spaceability are quite natural.

Large linear subspaces can often be found inside exotic subsets.

The new approach broadens the understanding of linear structures in vector spaces.

Abstract

The area of research called \textquotedblleft Lineability\textquotedblright% \ looks for linear structures inside exotic subsets of vector spaces. In the last decade lineability/spaceability has been investigated in rather general settings; for instance, Set Theory, Probability Theory, Functional Analysis, Measure Theory, etc. It is a common feeling that positive results on lineability/spaceability are quite natural (i.e., in general \textquotedblleft large\textquotedblright subspaces can be found inside exotic subsets of vector spaces, in quite different settings) and more restrictive approaches have been persecuted. In this paper we introduce and explore a new approach in this direction.