Shrinking and boundedly complete atomic decompositions in Fr\'echet spaces

TL;DR

This paper investigates atomic decompositions in Fréchet spaces, focusing on shrinking and boundedly complete types, their duality, and implications for reflexivity, with examples in function spaces.

Contribution

It introduces new definitions for shrinking and boundedly complete atomic decompositions in locally convex spaces and explores their duality and relation to reflexivity.

Findings

Characterization of when atomic decompositions are shrinking or boundedly complete

Duality relations between shrinking and boundedly complete decompositions

Concrete examples in function spaces illustrating these concepts

Abstract

We study atomic decompositions in Fr\'echet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete atomic decompositions on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional atomic decomposition is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete atomic decompositions in function spaces are also presented.