Proposition 8

Let E be a reflexive Banach space, B its unit ball, F a locally convex separated space and $u: E\rightarrow F$ a linear mapping. The following conditions are equivalent :

(i) The linear mapping $u$ is compact ;

(ii) The set $u(B)$ is a compact subset of F ;

(iii) The restriction of $u$ to B is a continuous mapping of the set B, endowed with the topology induced by $\sigma (E,E')$, into F ;

(iv) For every sequence $(x_n)_{n\in\mathbf{N}}$ of points of B which converges to 0 for the topology $\sigma (E,E')$, the sequence $(u(x_n))_{n\in\mathbf{N}}$ converges to 0 in F ;

(v) From every infinite sequence of points of $u(B)$, one can extract a sequence which converges in F ;

(vi) Every infinite sequence of points of $u(B)$ has an adherence value in F.

Since E is a reflexive space, B is a compact subset of $E_{\sigma}$ (TVS, IV, p. 17, prop. 6), hence (iii) implies (ii). Condition (ii) implies (i) by definition, and condition (i) implies (iii) by prop. 6. It is also elementary that (iii) implies (iv) and that (v) implies (vi).

We shall now prove that (iv) implies (v). Let $(x_n)$ be an infinite sequence of points of B. Since B is a compact subset of $E_{\sigma}$, there exists, by the theorem of $\breve{S}$mulian (TVS, IV, p. 36, th. 2), a sequence $(y_n)$, extracted from the sequence $(x_n)$, which converges in $E_{\sigma}$ to a limit $y$. The sequence $(y_n-y)$ is bounded in $E_{\sigma}$, hence in E (TVS, IV, p. 1, prop. 1); it is therefore contained in a homothetic set of B. If condition (iv) is satisfied, the sequence $(u(y_n-y))$ converges to 0 in F, and $(u(y_n))$, which is a sequence extracted from $(u(x_n))$, converges to $u(y)$. This proves that (iv) implies (v).

To conclude, let us prove that (vi) implies (ii). Let $j: F\rightarrow \widehat{F}$ be the canonical injection of F into its completion. Under hypothesis (vi), $u(B)$ is a precompact subset of F (TVS, IV, p. 32, prop. 1), hence $j(u(B))$ is a relatively compact subset of $\widehat{F}$ (TG, II, n$^{\circ}2$, p. 29) and $j\circ u$ is a compact linear mapping. By the equivalence already proved of conditions (i) and (ii), $j(u(B))$ is a compact subset of $\widehat{F}$. Consequently, $u(B)$ is a compact subset of F.