The incompleteness of weak duals

1. Incompleteness of weak duals of reasonable spaces 2. Appendix: locally-convex limits and colimits 3. Appendix: ubiquity of quasi-completeness The point here is to prove that the weak duals of reasonable topological vector spaces, such as infinite-dimensional Hilbert, Banach, or Fréchet spaces, are not complete. That is, in these weak duals there are Cauchy nets which do not converge. Happily, this incompleteness is not a fatal problem, because weak duals of Fréchet (and weak duals of strict inductive limits of Fréchet) are quasi-complete, and quasi-completeness suffices for most applications. For example, it is often observed that the space of distributions with the weak topology is sequentially complete, but sequential completeness is insufficient for many purposes, such as vector-valued integrals, whether Gelfand-Pettis (weak) or Bochner (strong). Fortunately, spaces of distributions are quasi-complete. It is an exercise to show that in metrizable spaces, sequential completeness implies completeness. Weak duals of infinite-dimensional topological vector spaces are rarely metrizable, so the distinctions among sequentially complete, quasi-complete, and complete are meaningful, and potentially significant. Again, sequential completeness is insufficient for many applications. One might worry that quasi-completeness of weak duals is not the best possible result, namely, one might imagine that some of these spaces are complete. The discussion here proves that completeness definitely does not hold. Again, fortunately, quasi-completeness is sufficient to prove existence of Gelfand-Pettis integrals of continuous compactly-supported vector-valued functions, and to prove that weakly holomorphic vector-valued functions are strongly holomorphic. See [Garrett], for example. The incompleteness (in the sense that not all Cauchy nets converge) of most weak duals has been known at least since [Grothendieck 1950], which gives a systematic analysis of completeness of various types of duals. This larger issue is systematically discussed in [Schaefer 1966/99], p. 147-8 and following. The quasi-completeness of weak duals and similar spaces is recalled in an appendix here.