Quasi-Compactness in Quasi-Banach Spaces

Raheam A Mansor Al-Saphory,Mahmood K Jasim

doi:10.24297/jam.v4i1.7227

Abstract

Quasi-compactness in a quasi-Banach space for the sequence space Lp, p< 0 < p <1 has been introduced based on the important extension of Milman's reverse Brunn-Minkowiski inequality by Bastero et al. in 1995. Moreover, Many interesting results connected with quasi-compactness and quasi-completeness in a quasi-normed space, Lp for 0 < p < 1 have been explored. Furthermore, we have shown that, the quasi-normed space under which condition is a quasi Banach space. Also, we have shown that the space if it is quasi-compact in quasi normed space then it is quasi Banach space and the converse is not true. Finally, a sufficient condition of the existence for a quasi-compact operator from Lp -> Lp has been presented and analyzed.

Highlights

Functional analysis is a scientific discipline of fairly recent origin
We have shown that, the quasi-normed space under which condition is a quasi Banach space
We have shown that the space if it is quasi-compact in quasi normed space it is a quasiBanach space and the converse is not true

Summary

Introduction

Functional analysis is a scientific discipline of fairly recent origin. It provides a power full tool to discover solution to problems occurring in pure, applied social sciences, for instance physics, engineering, medicine, agro-industries, ecology, economics and bio-mathematics [5, 16].One of the important notions in functional analysis is the concept of Banach space. Definition 2.13: A sequence { xn } in a q-normed space q X is called a q-bounded sequence if and only if there exists a positive real number M such that q || xn || M for all n N. Definition 2.16: A q-normed space, in which every q-Cauchy sequence is q-convergent, is called q-Banach space.

Results

Conclusion