Quasi-inner product spaces of quasi-Sobolev spaces and their completeness

Abstract

Sequences spaces , m , p have called quasi-Sobolev spaces were introduced by Jawad . K. Al-Delfi in 2013 [1]. In this paper , we deal with notion of quasi-inner product space by using concept of quasi-normed space which is generalized to normed space and given a relationship between pre-Hilbert space and a quasi-inner product space with important results and examples. Completeness properties in quasi-inner product space gives us concept of quasi-Hilbert space . We show that , not all quasi-Sobolev spaces , are quasi-Hilbert spaces. The best examples which are quasi-Hilbert spaces and Hilbert spaces are , where m . Finally, propositions, theorems and examples are our own unless otherwise referred.

Keywords

• quasi-Sobolev space, quasi-Banach space, Ga Ì‚teaux derivative , quasi-inner product space, quasi-Hilbert space. smooth quasi-Hilbert space.