Let ##(V, ||\cdot||)## be some finite-dimensional vector space over field ##\mathbb{F}## with ##\dim V = n##. Endowing this vector space with the metric topology, where the metric is induced by the norm, will ##V## become a topological vector space? It seems that this might be true, given that finite-dimensional vector spaces are nice e.g., all norms on a finite dimensional vector space are equivalent and ##V## is isomorphic ##\mathbb{F}^n##. In short, my question is, can all normed finite-dimensional vector spaces be made into a topological vector space by considering the metric topology?(adsbygoogle = window.adsbygoogle || []).push({});

Does anyone know if this is true? I would like to know before I attempt at proving it.

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# I Normed Vector Spaces and Topological Vector Spaces

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