2020/07/08 by Sánchez, Javier Cabello, Garmendia, José Navarro
#26E30 #46S10 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2007.04374
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 ℤ/3ℤ. Consequently, the results that have been proven for this class of spaces, like the Mazur-Ulam Theorem, are either trivial or empty statements.