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A note on Basis Problem in normed spaces

Vinicius Coelho, Joilson Ribeiro, Luciana Salgado

Abstract

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

A note on Basis Problem in normed spaces

Abstract

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

Paper Structure

This paper contains 17 sections, 17 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Bessaga-Pełczyński's Selection Principle
  4. Banach problem for normed spaces
  5. Spectral Theorem on inner product spaces
  6. Problems
  7. Organization of the text
  8. Auxilar Results
  9. Essential Schauder basis
  10. Proof of applications of main results
  11. Proof of Bessaga-Pełczyński's Selection Principle
  12. Proof of Banach problem for normed spaces
  13. Proof of Spectral Theorem
  14. Proofs of main results
  15. The Banach-Grunblum's criterion for normed spaces
  16. ...and 2 more sections

Key Result

Theorem 1.1

(Banach-Grunblum's Criterion) A sequence $(x_{n})_{n=1}^{\infty}$ of non null vector in a Banach space $X$ is a basic sequence if, and only if, there exists $M \geqslant 1$ such that for all sequence of scalar $(a_{n})_{n=1}^{\infty}$: whenever $n \geqslant m$.

Theorems & Definitions (40)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1.1
  • proof
  • Definition 4
  • Theorem A
  • Definition 5
  • Theorem B
  • Definition 6
  • ...and 30 more