Voidness of strict convexity in non-Archimedean fuzzy normed spaces

TL;DR

This paper demonstrates that non-Archimedean fuzzy normed spaces are essentially void of strict convexity, with only trivial or highly restricted cases existing, rendering many classical results trivial or inapplicable.

Contribution

It establishes that non-Archimedean fuzzy normed spaces lack strict convexity, showing the only nonzero case is a one-dimensional space over rac{1}{3}ield, thus challenging previous assumptions.

Findings

No strictly convex non-Archimedean fuzzy normed spaces exist.

The only nonzero strictly convex space is one-dimensional over rac{1}{3}ield.

Classical theorems like Mazur-Ulam are trivial or empty in this context.

Abstract

In this short note, we show by elementary computations that the notion of non-Archimedean fuzzy normed (and 2-normed) spaces is void. Namely, there are no strictly convex spaces at all --not even the zero-dimensional linear space. Before this, we also study the case of strictly convex non-Archimedean normed spaces; in this setting we see that the only nonzero linear space (defined over an arbitrary non-Archimedean field) that satisfies this property is the one-dimensional linear space over $\mathbb{Z}/3\mathbb{Z}$. Consequently, the results that have been proven for this class of spaces, like the Mazur-Ulam Theorem, are either trivial or empty statements.