Proposition 4

Suppose that $v$ is not improper and $K$ is complete. Let $E$ be a left topological vector space over $K$, which is Hausdorff and of finite dimension $n$. For every basis $(e_i)_{1 \leq i \leq n}$ of $E$ over $K$, the mapping $(a_i) \mapsto \sum_{i=1}^n a_i e_i$ of $K^n_s$ onto $E$ is a topological vector space isomorphism.

Taking account of Proposition 3 and its corollary, the proof is the same as that of Topological Vector Spaces, Chapter I, § 2, Theorem 2.