Complete topological vector space

In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point ${\displaystyle x}$ towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point ${\displaystyle x}$ towards which they all get closer" means that this Cauchy net or filter converges to ${\displaystyle x.}$ The notion of completeness for TVSs uses the theory of uniform spaces as a framework to generalize the notion of completeness for metric spaces. But unlike metric-completeness, TVS-completeness does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.

Completeness is an extremely important property for a topological vector space to possess. The notions of completeness for normed spaces and metrizable TVSs, which are commonly defined in terms of completeness of a particular norm or metric, can both be reduced down to this notion of TVS-completeness – a notion that is independent of any particular norm or metric. A metrizable topological vector space ${\displaystyle X}$ with a translation invariant metric^{[note 1]} ${\displaystyle d}$ is complete as a TVS if and only if ${\displaystyle (X,d)}$ is a complete metric space, which by definition means that every ${\displaystyle d}$-Cauchy sequence converges to some point in ${\displaystyle X.}$ Prominent examples of complete TVSs that are also metrizable include all F-spaces and consequently also all Fréchet spaces, Banach spaces, and Hilbert spaces. Prominent examples of complete TVS that are (typically) not metrizable include strict LF-spaces such as the space of test functions ${\displaystyle C_{c}^{\infty }(U)}$ with it canonical LF-topology, the strong dual space of any non-normable Fréchet space, as well as many other polar topologies on continuous dual space or other topologies on spaces of linear maps.

Explicitly, a topological vector spaces (TVS) is complete if every net, or equivalently, every filter, that is Cauchy with respect to the space's canonical uniformity necessarily converges to some point. Said differently, a TVS is complete if its canonical uniformity is a complete uniformity. The canonical uniformity on a TVS ${\displaystyle (X,\tau )}$ is the unique^{[note 2]} translation-invariant uniformity that induces on ${\displaystyle X}$ the topology ${\displaystyle \tau .}$ This notion of "TVS-completeness" depends only on vector subtraction and the topology of the TVS; consequently, it can be applied to all TVSs, including those whose topologies can not be defined in terms metrics or pseudometrics. A first-countable TVS is complete if and only if every Cauchy sequence (or equivalently, every elementary Cauchy filter) converges to some point.

Every topological vector space ${\displaystyle X,}$ even if it is not metrizable or not Hausdorff, has a completion, which by definition is a complete TVS ${\displaystyle C}$ into which ${\displaystyle X}$ can be TVS-embedded as a dense vector subspace. Moreover, every Hausdorff TVS has a Hausdorff completion, which is necessarily unique up to TVS-isomorphism. However, as discussed below, all TVSs have infinitely many non-Hausdorff completions that are not TVS-isomorphic to one another.
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