A note on Basis Problem in normed spaces

TL;DR

This paper establishes key criteria and principles in normed spaces, including the Banach-Grunblum criterion, Bessaga-Pełczyński selection principle, and applies these to spectral theory of compact self-adjoint operators.

Contribution

It proves the Banach-Grunblum criterion and Bessaga-Pełczyński selection principle specifically for normed spaces, extending their applicability.

Findings

Proved the Banach-Grunblum criterion for normed spaces.

Established the Bessaga-Pełczyński selection principle in this context.

Applied these results to the spectral theorem for compact self-adjoint operators.

Abstract

In this work, we prove the criterion of Banach-Grunblum and the principle of selection of Bessaga-Pe\l{}czy\'nski for normed spaces. As applications of these results, we show the Principle of Selection of Bessaga-Pe\l{}czy\'nski for normed spaces and the Spectral Theorem for compact self-adjoint operators on inner product spaces.