Zabreiko's lemma in 2-normed space and its applications

TL;DR

This paper extends Zabreiko's lemma to 2-Banach spaces and applies it to establish versions of the closed graph and open mapping theorems in this context.

Contribution

The paper introduces a generalized Zabreiko's lemma for 2-Banach spaces and demonstrates its use in proving fundamental theorems in functional analysis.

Findings

Zabreiko's lemma proven in 2-Banach spaces

Closed graph theorem established for 2-Banach spaces

Open mapping theorem proved for 2-Banach spaces

Abstract

We prove the Zabreiko's lemma in 2-Banach spaces. As an application we shall prove a version of the closed graph theorem and open mapping theorem.