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 towards which they all get closer. The notion of "points that get progressively closer" is made rigorous by or , which are generalizations of , while "point towards which they all get closer" means that this Cauchy net or filter converges to 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 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 with a translation invariant metric is complete as a TVS if and only if is a complete metric space, which by definition means that every -Cauchy sequence converges to some point in 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) metrizable include strict LF-spaces such as the space of test functions 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 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 is the unique translation-invariant uniformity that induces on the topology This notion of "TVS-completeness" depends 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 even if it is not metrizable or not Hausdorff, has a , which by definition is a complete TVS into which can be TVS-embedded as a dense vector subspace. Moreover, every Hausdorff TVS has a completion, which is necessarily unique up to TVS-isomorphism. However, as discussed below, all TVSs have infinitely many non-Hausdorff completions that are TVS-isomorphic to one another.
Definitions
This section summarizes the definition of a complete topological vector space (TVS) in terms of both nets and prefilters. Information about convergence of nets and filters, such as definitions and properties, can be found in the article about filters in topology.
Every topological vector space (TVS) is a commutative topological group with identity under addition and the canonical uniformity of a TVS is defined in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed.
Canonical uniformity
The of is the set
and for any the / is the set
where if then contains the diagonal
If is a symmetric set (that is, if ), then is , which by definition means that holds where and in addition, this symmetric set's with itself is:
If is any neighborhood basis at the origin in then the family of subsets of
is a prefilter on
If is the neighborhood filter at the origin in then forms a base of entourages for a uniform structure on that is considered canonical. Explicitly, by definition, is the filter on generated by the above prefilter:
where denotes the of in
The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. If is any neighborhood basis at the origin in then the filter on generated by the prefilter is equal to the canonical uniformity induced by
Cauchy net
The general theory of uniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net". For the canonical uniformity on these definitions reduce down to those given below.