Admissibility of Frechet spaces

TL;DR

This paper investigates the admissibility of certain Frechet spaces, extending previous results, and applies these findings to modular spaces with generalized F-norms, also exploring metric approximation properties in symmetric spaces.

Contribution

It generalizes the admissibility results of Frechet spaces and introduces a broader class of F-norms, including a linear version for symmetric spaces.

Findings

Admissibility of a large class of Frechet spaces established.

Generalization of Luxemburg F-norm to new F-norms.

Demonstration of metric approximation property in symmetric spaces.

Abstract

The aim of this paper is to to show the admissibility of some class of Frechet spaces (see Definition 2.3). In particular, this generalizes the main results of [3]. As an application, we show the admissibility of a large class modular spaces equipped with F-norms determined in Theorem 4.1. It is worth noticing that F-norms introduced in Theorem 4.1 generalize the classical Luxemburg F-norm. Also a linear version of admissibility (so called metric approximation property) for order continuous symmetric spaces will be demonstrated (see Theorem 5.1).