On absolutely representing families of subspaces in Banach spaces

TL;DR

This paper investigates conditions under which families of subspaces in Banach spaces can be considered absolutely representing, extending the concept from systems of elements to subspaces, with applications to families spanned by exponents.

Contribution

It introduces necessary and sufficient conditions for absolutely representing families of subspaces and explores their properties in Banach spaces, including specific examples involving exponent-spanned subspaces.

Findings

Established criteria for absolutely representing families of subspaces

Analyzed properties of these families in Banach spaces

Provided examples involving exponent-spanned subspaces

Abstract

An absolutely representing family of subspaces is a natural generalization of an absolutely representing system of subspaces and absolutely representing system (of elements). We obtain necessary an (or) sufficient conditions for a family of subspaces to be an absolutely representing family of subspaces and study properties of absolutely representing families of subspaces in Banach spaces. As an example, we study families of subspaces spanned by exponents.