Is a metrizable topological vector space normable?

Question

A tvs is metrizable iff it is $T_2$ and has a countable local base; while a tvs is normable iff it is $T_2$ and $0$ has a bounded convex neighbourhood. So any anybody give me an example of a metrizable tvs, which is not normable. Thanks.

Answer 1

The vector space $\mathcal C^\infty\bigl([0,1],\mathbf R\bigr)$ seems to be an example: you define its topology with the family of semi-norms: $$\lVert f\rVert_k=\sup_{x\in[0,1]}\bigl\lvert f^k(x)\bigr\rvert$$ It can be shown it is a Fréchet space for the topology defined by this family (hence it is metrisable), but its topology can't be defined by a single norm.

Answer 2

$\mathbb R^\mathbb N$, the space of real-valued sequences with the product topology. It has a tvs structure and is metrizable, but not normable. See Aliprantis–Border (2006, p. 207).