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A generalized notion of convergence of sequences of subspaces in an inner product space via ideals

Prasanta Malik, Saikat Das

Abstract

In this paper we introduce the notion of I-convergence of sequences of k-dimensional subspaces of an inner product space, where I is an ideal of subsets of N, the set of all natural numbers and k in N. We also study some basic properties of this notion.

A generalized notion of convergence of sequences of subspaces in an inner product space via ideals

Abstract

In this paper we introduce the notion of I-convergence of sequences of k-dimensional subspaces of an inner product space, where I is an ideal of subsets of N, the set of all natural numbers and k in N. We also study some basic properties of this notion.
Paper Structure (3 sections, 8 theorems, 29 equations)

This paper contains 3 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction and background:
  2. Basic Definitions and Notations:
  3. Main Results:

Key Result

Theorem 3.1

$\mathcal{I}-\lim_{n\rightarrow \infty}U_{n}= V$ if and only if $\mathcal{I}-\lim _{n\rightarrow \infty}\left\|u^{(n)}_{i}- P_{V}(u^{(n)}_{i})\right\|= 0, \forall~~~ i= 1, 2,...,k$.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • Theorem 3.1
  • proof
  • ...and 17 more