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arXiv 2007.043742020-07math.FA

Voidness of strict convexity in non-Archimedean fuzzy normed spaces

Javier Cabello Sánchez, José Navarro Garmendia

英文摘要：

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.

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