Remark

— We know (II, p. 48, prop. 7, (ii)) that $\iota$ is a topological vector space isomorphism from $E'_s/M^\circ$ onto $M'_s$ (weak duals). For the topology of compact convex convergence, prop. 10 shows that $\iota$ is an isomorphism from $E'_{cc}/M^\circ$ onto $M'_{cc}$ when $E$ is Hausdorff and $M$ closed in $E$. For the strong topologies, $\iota$ is a continuous mapping from $E'_b/M^\circ$ onto $M'_b$; it is an isomorphism if $E$ is a Banach space \* or if $E$ is semi-reflexive and $M$ is closed in $E$ (IV, p. 15) *, but this is not always so if $E$ is a Fréchet space (IV, p. 58, exerc. 5, c)).