New Classes of Generalized PN Spaces and Their Normability

In this paper, we obtain some properties of invertible operators; convex, balanced, absorbing sets; and D -boundedness in Šerstnev spaces. We prove that some PN spaces ( V , ν , τ , τ ∗ ), which are not Šerstnev spaces, in which the triangle function τ ∗ is not Archimedean can be endowed with a structure of a topological vector space, and we give suitable example to illustrate this result. Also, we show that the topological spaces obtained in such a manner are normable under certain given conditions: some examples are given.