Let $H$ be a Hilbert space, and let $X$ be a topological vector space.

• Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$?

Supposing that such an injective embedding $f : H \to X$ exists, consider the topological closure $F := \overline{fH}$ in $X$.

• When does the closure $F$ admit the structure of a Fréchet space, with $f : H \to F$ an isometric embedding?