Diametral dimensions of Frechet spaces

TL;DR

This paper introduces new variants of diametral dimensions in Frechet spaces to address longstanding conjectures and clarifies relationships between different topological invariants, enhancing understanding of nuclear Frechet spaces.

Contribution

It proposes novel variants of diametral dimensions and resolves an old conjecture for nuclear Frechet spaces, also clarifying relations between recent and classical invariants.

Findings

Solved an old conjecture for nuclear Frechet spaces.

Established relationships between new and classical invariants.

Enhanced understanding of topological invariants in Frechet spaces.

Abstract

The diametral dimension is an important topological invariant in the category of Frechet spaces which has been used, e.g., to distinguish types of Stein manifolds. We introduce variants of the classical definition in order to solve an old conjecture of Bessaga, Mityagin, Pelczynski, and Rolewicz at least for nuclear Frechet spaces. Moreover, we clarify the relation between an invariant recently introduced by Terzioglu and the by now classical condition $(\bar \Omega)$ of Vogt and Wagner.