Projective and Direct limits of Banach $G$ and tensor structures

TL;DR

This paper explores how projective and direct limits of Banach tensor structures can be endowed with Fréchet and convenient structures, respectively, and investigates adapted connections to $G$-structures with numerous examples.

Contribution

It introduces a framework for endowing limits of Banach tensor structures with suitable topologies and studies their connections to $G$-structures, supported by extensive examples.

Findings

Limits of Banach tensor structures can be given Fréchet or convenient structures.

Adapted connections to $G$-structures are constructed in both frameworks.

Numerous examples illustrate the theoretical developments.

Abstract

We endow projective (resp. direct) limits of Banach tensor structures with Fr\'{e}chet (resp. convenient) structures and study adapted connections to $G$-structures in both frameworks. This situation is illustrated by a lot of examples.