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Zabreiko's Lemma with Bicomplex and hyperbolic scalars and its applications

Akshay S. Rane, Mandar Thatte

Abstract

In this paper, we shall consider the notion of hyperbolic semi norm which on a module $X$ to set of all positive hyperbolic numbers. We shall prove the characterization of continuity of hyperbolic semi norm in this setup. We shall prove Zabreiko's lemma when $X$ is a F, $\mathbb{BC}$ module, where $\mathbb{BC}$ denotes the set of Bi complex numbers.(analogous to completeness). This lemma shall be used to prove the fundamental theorems of functional analysis like the Closed Graph Theorem, Open mapping Theorem, Uniform Boundedness principle.

Zabreiko's Lemma with Bicomplex and hyperbolic scalars and its applications

Abstract

In this paper, we shall consider the notion of hyperbolic semi norm which on a module to set of all positive hyperbolic numbers. We shall prove the characterization of continuity of hyperbolic semi norm in this setup. We shall prove Zabreiko's lemma when is a F, module, where denotes the set of Bi complex numbers.(analogous to completeness). This lemma shall be used to prove the fundamental theorems of functional analysis like the Closed Graph Theorem, Open mapping Theorem, Uniform Boundedness principle.
Paper Structure (4 sections, 10 theorems, 122 equations)

This paper contains 4 sections, 10 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Zabreiko's Lemma
  4. Application

Key Result

Theorem 2.1

Let X be a F-$\mathbb{B}\mathbb{C}$ module. Let $U_{n}$ be open dense subsets of X for n $\in \mathbb{N}$. Then $\bigcap_{n}U_{n}$ is dense in X.

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 11 more