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 towards which they all get closer to. 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 to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.
Banach spaceBanach–Alaoglu theoremBornological spaceBounded inverse theoremCanonical uniformityCanonical uniformity (topological vector space)Complemented subspaceComplete TVSComplete topological vector spaceDF-spaceDifferentiable vector-valued functions from Euclidean spaceDistribution (mathematics)Dual spaceFinal topologyFréchet spaceFréchet–Urysohn spaceFundamental theorem of Hilbert spacesInjective tensor productInner product spaceLB-spaceLocally complete topological vector spaceLocally convex topological vector spaceMetrizable topological vector spaceMontel spaceNuclear spaceOpen mapping theorem (functional analysis)Polar setPtak spaceRelatively compact subspaceSchwartz spaceSchwartz topological vector spaceSequence spaceSequential spaceSequentially completeTopological groupTopological homomorphismTopological vector spaceTotally bounded spaceUltrabornological space
Link from a Wikipage to another Wikipage
seeAlso
primaryTopic
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 towards which they all get closer to. 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 to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.
has abstract
In functional analysis and rel ...... TVS of -valued test functions.
@en
Wikipage page ID
53,695,856
page length (characters) of wiki page
Wikipage revision ID
1,023,043,504
Link from a Wikipage to another Wikipage
math statement
Let be metric on a vector s ...... space then is a complete-TVS.
@en
Let be metric on a vector sp ...... space then is a complete-TVS.
@en
Let be a complete TVS and let ...... rating family of seminorms for
@en
Let be a metrizable topologic ...... omorphism is also an isometry.
@en
Suppose is a complete Hausdor ...... uous linear extension to a map
@en
Suppose that and are Hausdor ...... ue TVS-isomorphism such that
@en
The topology of any TVS can be ...... is a base for this uniformity.
@en
name
Corollary
@en
Properties of Hausdorff completions
@en
Theorem
@en
note
Completions of quotients
@en
Existence and uniqueness of the canonical uniformity
@en
Klee
@en
Topology of a completion
@en
wikiPageUsesTemplate
subject
type
comment
In functional analysis and rel ...... e not metrizable or Hausdorff.
@en
label
Complete topological vector space
@en