Exercise 12, § 1

Let E be a Hausdorff topological vector space over a field K with a non-discrete valuation, and F a metrizable vector space over K. If u is a continuous linear mapping from E into F such that, for every bounded subset B of F, $u^{-1}(B)$ is bounded in E, show that u is an isomorphism from E onto a subspace of F.