Theorem 1 (Eberlein)

— Let E be a Hausdorff and quasi-complete locally convex space, $\mathscr{T}$ a topology on E which is compatible with the duality between E and E' and A a subset of E. For A to be relatively compact for $\mathscr{T}$, it is necessary and sufficient that every infinite sequence of points of A has a limit point in E for $\mathscr{T}$.

The condition stated is obviously necessary.

Suppose now that every infinite sequence of points of A has a limit point for $\mathscr{T}$, hence also for the coarser topology $\sigma(E, E')$. Then A is precompact for $\mathscr{T}$ (IV, p. 32, prop. 1); in order that A be relatively compact for $\mathscr{T}$, it is necessary and sufficient that it be so for $\sigma(E, E')$ (loc. cit.). Therefore it is enough to prove the theorem when $\mathscr{T}$ is the weakened topology $\sigma(E, E')$.

Let $\hat{E}$ denote the completion of E, which we shall identify as usual with a subspace of the algebraic dual ${E'}^*$ of E' (III, p. 21, th. 2). Let $E_\sigma$, $\hat{E}_\sigma$ and ${E_\sigma'}^*$ denote the spaces E, $\hat{E}$ and ${E'}^*$ endowed with the topologies $\sigma(E, E')$, $\sigma(\hat{E}, E')$ and $\sigma({E'}^*, E')$ respectively.

Let $(x'_i)_{i \in I}$ be a basis of the vector space E' over the field K. The mapping $f \mapsto (f(x'_i))_{i \in I}$ is a homeomorphism $\phi$ from ${E_\sigma'}^*$ onto $K^I$; for every $i \in I$, the image of A under the mapping $x'_i$ from E into K is relatively compact : for, K is metrizable and every infinite sequence of elements of $x'_i(A)$ has a limit point. It follows that $\phi(A)$ is relatively compact in $K^I$, hence that the closure $\overline{A}$ of A in ${E_\sigma'}^*$ is compact.

Next we shall prove that $\overline{A}$ is contained in $\hat{E}$. Let H be an equicontinuous subset of E'; let X be its closure for $\sigma(E', E)$; X is compact (III, p. 17, cor. 2). For every $x \in {E'}^*$, let $\phi_x$ be the restriction of $x' \mapsto \langle x, x' \rangle$ to X; let $\tilde{A} \subset \mathscr{C}_s(X)$ be the set of functions $\phi_x$ as x ranges over A. In view of the hypothesis on A, every infinite sequence of elements of $\tilde{A}$ has a limit point in $\mathscr{C}_s(X)$; by prop. 2 (IV, p. 33), the set $\tilde{A}$ is therefore relatively compact in $\mathscr{C}_s(X)$. It follows that for every $a \in \overline{A}$, the function $\phi_a$ on X is continuous. The inclusion $\overline{A} \subset \hat{E}$ then follows from th. 2 of III, p. 21.

Now we shall show that $\overline{A}$ is contained in E. Since A is precompact in $E_\sigma$ (IV, p. 32, prop. 1), it is bounded in $E_\sigma$ (III, p. 3, prop. 2), hence also in E (IV, p. 1, prop. 1). Let C be the closed convex balanced envelope of A in E. Then C is bounded since A is bounded, hence complete since E is quasi-complete. In other words, C is a convex and closed subset of $\hat{E}$, so also of $\hat{E}_\sigma$ (IV, p. 1, prop. 1). Since $A \subset C$ and the topology of $\hat{E}_\sigma$ is induced by that of ${E_\sigma'}^*$, we have $\overline{A} \subset C$, and hence $\overline{A} \subset E$.

Since the topology of $E_\sigma$ is induced by that of ${E'_\sigma}^*$, the subset $\overline{A}$ of $E_\sigma$ is compact, and th. 1 follows.