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 metricA metric on a vector space is said to be translation invariant if for all vectors A metric that is induced by a norm is always translation invariant. 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 uniqueCompleteness of normed spaces and metrizable TVSs are defined in terms of norms and metrics. In general, many different norms (for example, equivalent norms) and metrics may be used to determine completeness of such space. This stands in contrast to the uniqueness of this translation-invariant canonical uniformity. 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.
Suppose is a net in and is a net in The product becomes a directed set by declaring if and only if and Then
denotes the (Cartesian) , where in particular If then the image of this net under the vector addition map denotes the of these two nets:
and similarly their is defined to be the image of the product net under the vector subtraction map :
In particular, the notation denotes the -indexed net and not the -indexed net since using the latter as the definition would make the notation useless.
A net in a TVS is called a Cauchy net if
Explicitly, this means that for every neighborhood of in there exists some index such that for all indices that satisfy and It suffices to check any of these defining conditions for any given neighborhood basis of in A Cauchy sequence is a sequence that is also a Cauchy net.
If then in and so the continuity of the vector subtraction map which is defined by guarantees that in where and This proves that every convergent net is a Cauchy net. By definition, a space is called if the converse is also always true. That is, is complete if and only if the following holds: whenever is a net in then converges (to some point) in if and only if in A similar characterization of completeness holds if filters and prefilters are used instead of nets.
A series is called a (respectively, a ) if the sequence of partial sums is a Cauchy sequence (respectively, a convergent sequence). Every convergent series is necessarily a Cauchy series. In a complete TVS, every Cauchy series is necessarily a convergent series.
Cauchy filter and Cauchy prefilter
A prefilter on a topological vector space is called a Cauchy prefilter if it satisfies any of the following equivalent conditions: in The family is a prefilter. Explicitly, means that for every neighborhood of the origin in there exist such that in The family is a prefilter equivalent to (equivalence means these prefilters generate the same filter on ). Explicitly, means that for every neighborhood of the origin in there exists some such that For every neighborhood of the origin in contains some -small set (that is, there exists some such that ). A subset is called -small or if For every neighborhood of the origin in there exists some and some such that This statement remains true if "" is replaced with "" Every neighborhood of the origin in contains some subset of the form where and It suffices to check any of the above conditions for any given neighborhood basis of in A Cauchy filter is a Cauchy prefilter that is also a filter on
If is a prefilter on a topological vector space and if then in if and only if and is Cauchy.
Complete subset
For any a prefilter is necessarily a subset of ; that is,
A subset of a TVS is called a if it satisfies any of the following equivalent conditions: Every Cauchy prefilter on converges to at least one point of If is Hausdorff then every prefilter on will converge to at most one point of But if is not Hausdorff then a prefilter may converge to multiple points in The same is true for nets. Every Cauchy net in converges to at least one point of is a complete uniform space (under the point-set topology definition of "complete uniform space") when is endowed with the uniformity induced on it by the canonical uniformity of
The subset is called a if every Cauchy sequence in (or equivalently, every elementary Cauchy filter/prefilter on ) converges to at least one point of
Importantly, : If is not Hausdorff and if every Cauchy prefilter on converges to some point of then will be complete even if some or all Cauchy prefilters on converge to points(s) in In short, there is no requirement that these Cauchy prefilters on converge to points in The same can be said of the convergence of Cauchy nets in
As a consequence, if a TVS is Hausdorff then every subset of the closure of in is complete because it is compact and every compact set is necessarily complete. In particular, if is a proper subset, such as for example, then would be complete even though Cauchy net in (and also every Cauchy prefilter on ) converges to point in including those points in that do not belong to This example also shows that complete subsets (and indeed, even compact subsets) of a non-Hausdorff TVS may fail to be closed. For example, if then if and only if is closed in
Complete topological vector space
A topological vector space is called a if any of the following equivalent conditions are satisfied:
is a complete uniform space when it is endowed with its canonical uniformity. In the general theory of uniform spaces, a uniform space is called a complete uniform space if each Cauchy filter on converges to some point of in the topology induced by the uniformity. When is a TVS, the topology induced by the canonical uniformity is equal to 's given topology (so convergence in this induced topology is just the usual convergence in ). is a complete subset of itself. There exists a neighborhood of the origin in that is also a complete subset of This implies that every locally compact TVS is complete (even if the TVS is not Hausdorff). Every Cauchy prefilter on converges in to at least one point of If is Hausdorff then every prefilter on will converge to at most one point of But if is not Hausdorff then a prefilter may converge to multiple points in The same is true for nets. Every Cauchy filter on converges in to at least one point of Every Cauchy net in converges in to at least one point of
where if in addition is pseudometrizable or metrizable (for example, a normed space) then this list can be extended to include:
is sequentially complete.
A topological vector space is if any of the following equivalent conditions are satisfied: is a sequentially complete subset of itself. Every Cauchy sequence in converges in to at least one point of Every elementary Cauchy prefilter on converges in to at least one point of Every elementary Cauchy filter on converges in to at least one point of
Uniqueness of the canonical uniformity
The existence of the canonical uniformity was demonstrated above by defining it. The theorem below establishes that the canonical uniformity of any TVS is the only uniformity on that is both (1) translation invariant, and (2) generates on the topology
This section is dedicated to explaining the precise meanings of the terms involved in this uniqueness statement.
Uniform spaces and translation-invariant uniformities
For any subsets let
and let
A non-empty family is called a or a if is a prefilter on satisfying all of the following conditions: Every set in contains the diagonal of as a subset; that is, for every Said differently, the prefilter is on For every there exists some such that For every there exists some such that