About weak integrals: Appendix of Folland's book "A course in abstract harmonic analysis" Consider the following fragment from Folland's book "A course in abstract harmonic analysis":

All integrals are here to interpreted in the weak sense (see p285 in Folland's book). Why is the assumption that $\mathcal{V}$ is a Fréchet necessary in A.20 Theorem? I checked the reference given to Rudin's book "Functional analysis" and nothing there suggests that $\mathcal{V}$ should be Fréchet. In fact, it seems to me that we can safely assume that $\mathcal{V}$ is a locally convex (Hausdorff) space in A.20 Theorem. Is this correct?
 A: Among other true things that may be of interest in this context: compactly-supported continuous functions $f$ taking values in a locally-convex, quasi-complete topological vector space have Gelfand-Pettis integrals.
("Quasi-complete" includes Hilbert, Banach, Frechet, their weak duals, and almost any TVS of practical use that could reasonably be expected to be "complete" in some sense. Attempting to require "full completeness" (convergence of arbitrary Cauchy nets) already fails for the weak dual of infinite-dimensional Hilbert spaces... )
Indeed, some completeness notion is required. Hausdorffness (which I myself would take as a part of the expectation/definition of a "topological vector space") is certainly not enough.
Requiring "Frechet" is too much, though the completeness notion there is relatively elementary, being the metric space notion. I think that's all that Rudin really treats in this regard, though it becomes clear that that restriction is not needed. It has always been a mild mystery to me that Rudin did not prove the quasi-completeness of spaces of distributions (or tempered), to be able to talk about nice integrals of them, nor a Cauchy-Goursat-Schwartz-Grothendieck complex analysis for distribution-valued functions...
EDIT: in response to comments (especially @JochenWengenroth's very helpful observations...) first, yes, Rudin only assumes that convex hulls of compacts are compact. He only proves this in Frechet, but it is true (due to Bourbaki et al, for example) in quasi-complete, locally convex spaces. (Jacquet pointed out to me some decades ago that Bourbaki's "Integration Theory" does foreground these ideas. I had not been aware, though I was aware of Schwartz' and Grothendieck's work in functional analysis...)
And, yes, I meant "weak-*" topology on the dual, not weak topology on the original. While we're here, a more general result is that $\mathrm{Hom}(X,Y)$ is (locally convex and) quasi-complete for $X$ LF-spaces (strict limit of Frechet) and quasi-complete (locally convex) $Y$, with topologies ranging from the finite-to-open to compact-to-open to bounded-to-open. I executed this exercise at https://www-users.cse.umn.edu/~garrett/m/v/QC_theorem.pdf
