Corollary 2

Let E be a seminormed space and F a Banach space of countable type. Let $u$ be a continuous linear mapping of E into F. Suppose that for every sequence $(y_n)$ of elements of $F'$ converging weakly to 0, the sequence $(^tu(y_n))$ converges strongly to 0 in $E'$. Then the linear mapping $u$ is compact.

Let us denote by $B'$ the unit ball of $F'$, endowed with the topology induced by the topology $\sigma (F',F)$. Since F is a Banach space of countable type, $B'$ is a compact metrisable space (EVT, III, p. 19, cor. 2). Under the assumption made, the restriction of $^tu$ to $B'$ is a continuous mapping of $B'$ into $E'$ and $^tu(B')$ is a compact subset of $E'$. By cor. 1, the linear mapping $u$ is compact.