Proposition 5

Let $u$ be a continuous linear mapping from $E_1$ into $E_2$.

(i) If $E_1$ and $E_2$ are Hausdorff then $u$ is injective if and only if the image of $^t u$ is dense in $E'_1$ for the weak topology $\sigma(E'_1 : E_1)$.

(ii) For $^t u$ to be injective it is necessary and sufficient that the image of $u$ is dense in $E_2$.

A vector subspace of $E_2$ is dense for the initial topology if and only if it is dense for the weakened topology (IV, p. 4, prop. 2). Prop. 5 then follows from II, p. 47, cor. 2.