Invariance of Fr\'echet Frames under Perturbation

TL;DR

This paper introduces the concept of invariance of Fréchet frames under perturbation, demonstrating their stability and the existence of series expansions for elements in Fréchet spaces.

Contribution

It establishes the invariance property of Fréchet frames under perturbation and proves the existence of series expansions for all elements in Fréchet spaces.

Findings

Fréchet frames are invariant under perturbation.

Every element in a Fréchet space has a series expansion.

The paper extends frame theory to Fréchet spaces.

Abstract

Motivating the perturbations of frames in Hilbert and Banach spaces, in this paper we introduce the invariance of Fr\'echet frames under perturbation. Also we show that for any Fr\'echet spaces, there is a Fr\'echet frame and any element has a series expansion.