Lemma 1

Let E and F be two locally convex metrizable spaces, and u be a continuous linear mapping from $E'_b$ into F. Then there exists a neighbourhood U of 0 in $E'_b$ whose image under u is bounded in F.

Let $(U_n)_{n \in \mathbf{N}}$ (resp. $(V_n)_{n \in \mathbf{N}}$) be a fundamental system of neighbourhoods of 0 in E (resp. F). We assume that the sets $U_n$ are balanced and form a decreasing sequence. Since u is continuous, for every $n \in \mathbf{N}$, there exists a bounded set $B_n$ in E such that $u(B_n^\circ) \subset V_n$. Since $B_n$ is bounded, there exists a real number $\lambda_n > 0$ such that $\lambda_n B_n \subset U_n$. Put $B = \bigcup_{n \in \mathbf{N}} \lambda_n B_n$.

We shall prove that the set B is bounded in E, in other words that for every integer $m \geqslant 0$, there exists a real number $\mu > 0$ such that $\mu B \subset U_m$. Since the sets $B_n$ are bounded, there exists a real number $\mu$ such that $0 < \mu \leqslant 1$ and such that $\mu (\lambda_n B_n) \subset U_m$ for $0 \leqslant n \leqslant m$; we have also $\lambda_n B_n \subset U_n \subset U_m$ if $n > m$; hence $\mu B \subset U_m$ since $U_m$ is balanced.

Let U be the polar of B in $E'_b$. This is a neighbourhood of 0 in $E'_b$ and we have $\lambda_n B^\circ \subset B_n^\circ$, hence $\lambda_n u(U) \subset V_n$ for all $n \in \mathbf{N}$. Consequently $u(U)$ is bounded in F.