Some new results of completion t^ω- Normed approach space

In this paper , the completion of t^ω-approach spaces, isometric in t^ω-approach space and equivalent sequences are defined. Every t^ω-approach normed space can be embedded in t^ω-approach Banach space is proved, χ ̃ and an isometry ψ from χ onto the subspace F of χ ̃ which is dense in χ ̃ are introduced, as well as, the completion is shown an uniquely up to an isometry. In addition, to what is mentioned above, some essential definitions, examples, and important theorems are included to illustrate some of our work

New Kind of Banach Algebra Via Proximit structure

This article introduces the concept of Banach proximit algebra and examines several results in this new context. We give new definitions for the terms "proximit algebra," "commutative proximit algebra," "identity proximit algebra," and "normed proximit algebra." However, we introduce the concepts of subproximit algebra, proximit homomorphism, and proximit isomorphism of Banach proximit algebras in a natural way. Finally, we offer a few intriguing possibilities to support our arguments.