Lemma 1

Let E and F be topological vector spaces, $u$ a continuous linear mapping of E into F and U a neighbourhood of 0 in E. Suppose that $u$ induces a homeomorphism of U onto a closed subset of F. Then the image I of $u$ is closed and $u$ induces a homeomorphism of E onto I.

The set $u(U)$ is a neighbourhood of 0 in I; it is closed in F, hence the subgroup I of F is locally closed at 0, and consequently it is closed (TG, III, p. 7, prop. 4).

Since Ker($u$) meets U only at 0, we have Ker($u$) $=\{0\}$ and the mapping $u$ is injective. Let V be a closed neighbourhood of 0 in E contained in U. Since $u(\mathring{V})$ is a neighbourhood of 0 in $u$(V), there exists a balanced neighbourhood W of 0 in F such that $u(V)\cap W\subset u(\mathring{V})$. The set $\overset{-1}{u}(W)$ is balanced, hence connected. It contains 0 and is contained in $\mathring{V}\cup (E$ - V). Since $\mathring{V}$ and E- V are disjoint open subsets of E, the set $\overset{-1}{u}(W)$ is contained in $\mathring{V}$, whence $W\cap I\subset u(\mathring{V})$. Consequently $u(\mathring{V})$ is a neighbourhood of 0 in I. This implies that $u$ induces a homeomorphism of E onto I.