Analytic sheaves in Banach spaces

TL;DR

This paper introduces cohesive sheaves in Banach spaces, extending the concept of coherence to infinite dimensions, and proves analogs of Cartan's theorems A and B under certain conditions.

Contribution

It defines cohesive sheaves in Banach spaces and establishes foundational theorems analogous to finite-dimensional complex analysis.

Findings

Cohesive sheaves generalize coherence in Banach spaces.

Cartan's Theorems A and B are proved for cohesive sheaves.

Results hold for Banach spaces with an unconditional basis.

Abstract

We introduce a class of analytic sheaves in a Banach space X, that we call cohesive sheaves. Cohesion is meant to generalize the notion of coherence from finite dimensional analysis. Accordingly, we prove the analog of Cartan's Theorems A and B for cohesive sheaves over pseudoconvex open subsets of X, provided X has an unconditional basis.