Remark

— Let F be a barreled space and $\mathscr{T}$ a locally convex topology on $F'$ intermediate to $\mathscr{T}_s$ and $\mathscr{T}_c$. Every subset of $F'$ compact for $\mathscr{T}$ is compact for the coarser topology $\mathscr{T}_s$. Conversely, let K be a subset of $F'$ compact for $\mathscr{T}_s$. Since F is barreled, K is equicontinuous (TVS, III, §4, No. 2, Th. 1); but by Ascoli’s theorem, every equicontinuous subset of $F'$ is relatively compact for $\mathscr{T}_c$ and a fortiori for $\mathscr{T}$, therefore K is contained in a subset of $F'$ compact for $\mathscr{T}$. It is not difficult to infer from this that the identity mapping of $F'_\mathscr{T}$ onto $F'_{\mathscr{T}_s}$ defines a bijection between the sets of measures on these two spaces.