Corollary 3

— Let $G_1$ and $G_2$ be two vector subspaces of $F^*$, in duality with $F$ (for the restriction of the canonical bilinear form). Then $\sigma(F, G_2)$ is finer than $\sigma(F, G_1)$ if and only if $G_1 \subset G_2$.

The condition is obviously sufficient; conversely, if $\sigma(F, G_2)$ is finer than $\sigma(F, G_1)$, then every linear form that is continuous for $\sigma(F, G_1)$ is also continuous for $\sigma(F, G_2)$, hence $G_1 \subset G_2$ by prop. 3.