Proposition 4

— Let E be a reflexive Fréchet space. The strong dual $E'_b$ of E is the inductive limit of a sequence of Banach spaces.

Let $(V_n)_{n \in \mathbf{N}}$ be a decreasing sequence of convex, balanced and closed neighbourhoods of 0 in E, such that every neighbourhood of 0 in E contains one of the $V_n$. Let $A_n$ be the polar of $V_n$ in $E'$. Then $A_n$ is convex, balanced and compact for $\sigma(E', E)$; by III, p. 8, corollary the space $E'_{A_n}$ is a Banach space. We shall prove that $E'_b$ is the inductive limit of the spaces $E'_{A_n}$; in other words, that every convex and balanced subset U of $E'$ which absorbs each of the $A_n$ is a neighbourhood of 0 in $E'_b$. For every $n \in \mathbf{N}$, choose a real number $\lambda_n > 0$ such that $\lambda_n A_n \subset U$. Let $B_n$ be the convex envelope of the set $\bigcup_{0 \leq i \leq n} \lambda_i A_i$; put $V = \bigcup_n B_n$, then $V \subset U$. For every $n \in \mathbf{N}$, the set $B_n$ is convex, balanced and compact for $\sigma(E', E)$ (II, p. 14, prop. 15).

Now we shall show that $\frac{1}{2} V^{oo} \subset V$. Let $x \in E'_b - V$; for every $n \in \mathbf{N}$, we have $x \notin B_n$, and since $B_n$ is closed for $\sigma(E', E)$ there exists an element $y_n$ in $B_n^\circ$ such that \langle y_n, x \rangle = 1 (\text{II, p. 38, prop. 4}). Since E is reflexive, every bounded subset of E is relatively compact for $\sigma(E, E')$ (IV, p. 16, th. 2). By the definition of $B_n$, we have

hence the sequence $(y_n)$ is bounded. Let $y$ be a limit point of $(y_n)$ for the topology $\sigma(E, E')$. We have $y \in V^\circ = \bigcap_n B_n^\circ$ and $\langle y, x \rangle = 1$. Hence $x \notin \frac{1}{2}V^{\circ\circ}$, and so we have the inclusion $\frac{1}{2}V^{\circ\circ} \subset V$ and a fortiori, $\frac{1}{2}V^{\circ\circ} \subset U$.

Since every bounded subset of $E'_b$ is contained in one of the sets $A_n$, the set $V = \bigcup_n B_n$ absorbs every bounded subset of $E'_b$. Consequently, $V^\circ$ is bounded in E, hence $\frac{1}{2}V^{\circ\circ}$ is a neighbourhood of 0 in $E'_b$. A fortiori, U is a neighbourhood of in $E'_b$.