[ φ,p ]-normed spaces and Menger [ φ,p ]-normed spaces

Abstract

In this paper, we introduce the notion of [φ, p]-normed spaces, following the concept of ω-norms which was presented by Singh, and study the Aleksandrov problem in [φ, p]-normed spaces (0 < p ≤ 1). On the other hand, we introduce the concept of Menger [φ, p]-normed spaces, which includes the Menger φ-normed spaces defined by Golet as a special case, and present the topological properties of Menger [φ, p]-normed spaces with some results of profile function.