. . . . . . . . . . . . . . "93069"^^ . . . . "Let be a complete TVS and let be a dense vector subspace of \nIf is any neighborhood base of the origin in then the set\n\nis a neighborhood of the origin in the completion of \n\nIf is locally convex and is a family of continuous seminorms on that generate the topology of then the family of all continuous extensions to of all members of is a generating family of seminorms for"@en . . . "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 Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, 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 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 \u2013 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 Fr\u00E9chet 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 with it canonical LF-topology, the strong dual space of any non-normable Fr\u00E9chet 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 is the unique translation-invariant uniformity that induces on the topology 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 even if it is not metrizable or not Hausdorff, has a completion, which by definition is a complete TVS into which 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."@en . "Properties of Hausdorff completions"@en . . . . . . . . . . . . . . . . . . . . . . "Theorem"@en . . . . . . "53695856"^^ . . "Topology of a completion"@en . . "1124136590"^^ . . . . . . . . . . . . . . . . . . . . . . . . . "Corollary"@en . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "Existence and uniqueness of the canonical uniformity"@en . . . . . . . . . . . . . . . . . . . "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 Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, 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 all TVSs, including those that ar"@en . . . . . . . . . . . . . . . . . "Suppose that and are Hausdorff TVSs with complete. Suppose that is a TVS-embedding onto a dense vector subspace of Then\n\n:Universal property: for every continuous linear map into a complete Hausdorff TVS there exists a unique continuous linear map such that \n\nIf is a TVS embedding onto a dense vector subspace of a complete Hausdorff TVS having the above universal property, then there exists a unique TVS-isomorphism such that"@en . . "Completions of quotients"@en . . . . . . . . . . "Complete topological vector space"@en . . . . . . "Let be metric on a vector space such that the topology induced by on makes into a topological vector space. If is a complete metric space then is a complete-TVS."@en . . . . . . . . . "Klee"@en . . . . "Let be a metrizable topological vector space and let be a closed vector subspace of Suppose that is a completion of Then the completion of is TVS-isomorphic to If in addition is a normed space, then this TVS-isomorphism is also an isometry."@en . . . . "Suppose is a complete Hausdorff TVS and is a dense vector subspace of Then every continuous linear map into a complete Hausdorff TVS has a unique continuous linear extension to a map"@en . . . . . . . . . . . . . . . "Let be metric on a vector space such that the topology induced by on makes into a topological vector space. If is a complete metric space then is a complete-TVS."@en . . . . . . "The topology of any TVS can be derived from a unique translation-invariant uniformity. If is any neighborhood base of the origin, then the family is a base for this uniformity."@en . . . . .