# Condition under which a locally convex topological vector space becomes a normed linear space

Is this true that a locally convex topological (Hausdorff) vector space becomes a normed space when its local base has only one element, so only one Minkowski functional and so only one seminorm and hence norm.

This result was proved in 1935 by A. Kolmogoroff: The topology of a locally convex topological vector space is generated by a norm iff there exists a bounded neighborhood of $0$.