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On Schauder Bases in Hilbert Space

Oleg Zubelevich

Abstract

In this short note we present a far generalization of the following very well-known assertion: assume that we have two orthonormal sequences in a Hilbert space and these sequences are quadratically close to each other. Then if one of these sequences is a basis in the Hilbert space then so is the other one.

On Schauder Bases in Hilbert Space

Abstract

In this short note we present a far generalization of the following very well-known assertion: assume that we have two orthonormal sequences in a Hilbert space and these sequences are quadratically close to each other. Then if one of these sequences is a basis in the Hilbert space then so is the other one.

Paper Structure

This paper contains 3 sections, 3 theorems, 21 equations.

Table of Contents

  1. The main theorem
  2. A note on Banach spaces
  3. Proof of theorem \ref{['sdgftt']}

Key Result

Theorem 1

Suppose that a sequence of vectors satisfy the following pair of conditions: 1) the series is convergent: 2) the equality $\sum_{i=1}^\infty\lambda_if_i=0,\quad \lambda_i\in\mathbb{R}$ implies (This property of system (bi) is called the $\omega-$independence.) Then the sequence (bi) is a Schauder basis of $H$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Lemma 1