The Dvoretsky-Rogers Theorem for vector valued integrals on function spaces

TL;DR

This paper extends the Dvoretsky-Rogers Theorem to vector valued integrals on Banach function spaces, revealing conditions under which summability implies finite dimensionality and establishing new convergence results.

Contribution

It introduces a Dvoretsky-Rogers type theorem for q-summing operators in the context of vector valued integrals, relaxing classical assumptions and exploring infinite dimensional cases.

Findings

Summability of the identity map does not imply finite dimensionality.

Local compactness assumptions are necessary for the theorem.

Convergence of integrals can be equivalent to norm convergence in certain infinite dimensional spaces.

Abstract

We show a Dvoretsky-Rogers type Theorem for the adapted version of the $q$-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guaranty that the space has to be finite dimensional, contrarily to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals ---our vector valued version of convergence in the weak topology--- is equivalent to the convergence with respect to the norm. Examples and applications are also given.