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 these points will necessarily converge to some point in the space. The notion of "points that get progressively closer" is made rigorous by Cauchy nets and Cauchy filters, which generalize Cauchy sequences. 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, even those that are not metrizable or Hausdorff. Completeness is an extremely important property for a topological vector space to possess. Prominent examples of complete TVSs that are also metrizable include all Fréchet spaces, Banach spaces, and Hilbert spaces. Prominent examples of complete TVS that are (typically) not metrizable include strict LF-spaces and nuclear spaces such as the Schwartz space of smooth functions and also the spaces of distributions and test functions.

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 (X, τ) is the unique[note 1] translation-invariant uniformity that induces on X the topology τ. This notion of "TVS-completeness" depends only on vector subtraction and the topology of the TVS and so can be applied to all TVSs, including those whose topologies can not be defined in terms metrics (or pseudometrics).

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 X with a translation invariant metric d is complete as a TVS if and only if (X, d) is a complete metric space, meaning that every Cauchy sequence converges to some point.

All topological vector spaces, even those that are not metrizable or Hausdorff, have a completion. Moreover, every Hausdorff TVS has a Hausdorff completion that is unique up to TVS-isomorphism. However, as discussed below, all TVSs (even those that are complete, Hausdorff, and/or metrizable) have infinitely many non-Hausdorff completions that are not TVS-isomorphic to one another.

Definitions
Main articles: Net (mathematics) and Filters in topology

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 spaces (TVS) is a commutative topological group with identity under addition and the canonical uniformity of a TVS is defined entirely in terms of subtraction (and thus addition); scalar multiplication is not involved and no additional structure is needed.

Canonical uniformity

The diagonal of X is the family[1] ΔX  ≝  { (x, x) : x ∈ X }. For any N ⊆ X, the canonical entourage/vicinity around N is the set

ΔX(N)  ≝  { (x, y) ∈ X × X  :  x - y ∈ N }  =  ∪y ∈ X [(y + N) × { y }]  =  ΔX + (N × { 0 })

where if 0 ∈ N then ΔX(N) contains the diagonal ΔX  ≝  { (x, x) : x ∈ X  }  =  ΔX({0}).

If N is symmetric (i.e. if - N = N), then ΔX(N) is symmetric, which by definition means that (ΔX(N))op  ≝  { (y, x)  :  (x, y) ∈ ΔX(N) }  =  ΔX(N), and in addition, this symmetric set's composition with itself is:

ΔX(N) ∘ ΔX(N)  ≝  { (x, z) ∈ X × X  :  ∃ y ∈ X  such that  x, z ∈ y + N }  =  ∪y ∈ X [(y + N) × (y + N)]  =  ΔX + (N × N).

If 𝒮 is any neighborhood basis at the origin in (X, τ) then the family of subsets of X × X :

ℬ𝒮  ≝  { ΔX(N)  :  N ∈ 𝒮 }

is a prefilter on X × X. If 𝒩τ(0) is the neighborhood filter at the origin in (X, τ) then ℬ𝒩τ(0) forms a base of entourages for a uniform structure on X that is considered canonical.[2] Explicitly, by definition, the canonical uniformity on X induced by (X, τ)[2] is the filter 𝒰τ on X × X generated by the above prefilter:

𝒰τ  ≝  ℬ𝒩τ(0)↑X×X  ≝  { S ⊆ X × X  :  N ∈ 𝒩τ(0) and ΔX(N) ⊆ S }

where ℬ𝒩τ(0)↑X×X denotes the upward closure of ℬτ in X × X.

If 𝒮 is any neighborhood basis at the origin in (X, τ) then the filter on X × X generated by the prefilter ℬ𝒮 is equal to the canonical uniformity 𝒰τ induced by (X, τ).

Uniform continuity

Let X and Y be TVSs, D ⊆ X, and f : D → Y be a map. Then f : D → Y is uniformly continuous if for every neighborhood U of the origin in X, there exists a neighborhood V of the origin in Y such that for all  x, y ∈ D,  if  y - x ∈ U  then  f (y) - f (x) ∈ V.

Cauchy net
See also: Net (mathematics)

The general theory of uniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net". For the canonical uniformity on X, these definitions reduces down to the definition described below.

Suppose x• = (xi)i ∈ I is a net in X and y• = (yi)j ∈ J is a net in Y. The product I × J becomes a directed set by declaring (i, j) ≤ (i2, j2) if and only if i ≤ i2 and j ≤ j2. Then

x• × y•  ≝  (xi, yj)(i, j) ∈ I×J

denotes the product net. If X = Y then the image of this net under the vector addition map X × X → X denotes the sum of these two nets:[3]

x• + y•  ≝  ( xi + yj )(i, j) ∈ I×J

and similarly their difference is defined to be the image of the product net under the vector subtraction map:

x• - y•  ≝  ( xi - yj )(i, j) ∈ I×J.

A net x• = (xi)i ∈ I in a TVS X is called a Cauchy net[4] if

( xi - xj )(i, j) ∈ I×I  →  0  in  X

or equivalently, if for every neighborhood N of 0 in X, there exists some i0 ∈ I such that xi - xj ∈ N for all i, j ≥ i0 with i, j ∈ I. A Cauchy sequence is a Cauchy net that is a sequence. It suffices to check any of these defining conditions for any given neighborhood basis of 0 in X.

Cauchy filter and Cauchy prefilter
See also: Filters in topology

A prefilter ℬ on an TVS (X, τ) called a Cauchy prefilter[5] if it satisfies any of the following equivalent conditions:

  1. ℬ  -  ℬ  →  0  in X, where  ℬ  -  ℬ  ≝  { B  -  C  :  B, C ∈ ℬ } is a prefilter.
  2. { B  -  B  :  B ∈ ℬ }  →  0  in X, where  { B  -  B  :  B ∈ ℬ } is a prefilter equivalent to ℬ  -  ℬ.
  3. For every neighborhood N of 0 in X, ℬ contains some N-small set (i.e. there exists some B ∈ ℬ such that B - B ⊆ N).[6]
    • If B is a subset of a TVS X and N is a set containing 0, then B is N-small or small of order N[5] if B - B ⊆ N.
  4. For every neighborhood N of 0 in X, there exists some B ∈ ℬ and some x ∈ X such that B ⊆ x + N.[5]

It suffices to check any of the above conditions for any given neighborhood basis of 0 in X.

If ℬ is a prefilter on a TVS X and x ∈ X then ℬ → x in X if and only if x ∈ cl ℬ and ℬ is Cauchy.[3]

Complete subset
Main article: Complete uniform space

For any S ⊆ X, a prefilter 𝒞 on S is necessarily a subset of ℘(S); that is, 𝒞 ⊆ ℘(S).

A subset S of a TVS (X, τ) is called complete if it satisfies any of the following equivalent conditions:

  1. Every Cauchy prefilter 𝒞 ⊆ ℘(S) on S converges to at least one point of S.
    • If X is Hausdorff then every prefilter on S will converge to at most one point of X. But if X is not Hausdorff then a prefilter may converge to multiple points in X. The same is true for nets.
  2. Every Cauchy net in S converges to at least one point of S.
  3. S is a complete uniform space (under the point-set topology definition of "complete uniform space") when S is endowed with the uniformity induced on it by the canonical uniformity of X.

The subset S is called sequentially complete if every Cauchy sequence in S (or equivalently, every elementary Cauchy filter/prefilter on S) converges to at least one point of S.

Importantly, convergence outside of S is allowed: If X is not Hausdorff and if every Cauchy prefilter on S converges to some point of S, then S will be complete even if some or all Cauchy prefilters on S also converge to points(s) in X ∖ S. In short, there is no requirement that these Cauchy prefilters on S converge only to points in S. The same can be said of the convergence of Cauchy nets in S.

As a consequence, if a TVS X is not Hausdorff then every subset of the closure of { 0 } in X is complete because it is compact and every compact set is necessarily complete. In particular, if ∅ ≠ S ⊆ ClX { 0 } is a proper subset, such as S = { 0 } for example, then S would be complete even though every Cauchy net in S (and also every Cauchy prefilter on S) converges to every point in ClX { 0  }, including those points in ClX { 0 } that do not belong to S. 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 ∅ ≠ S ⊆ ClX { 0 } then S = ClX { 0 } if and only if S is closed in X.

Complete topological vector space

A topological vector space (X, τ) is called complete if any of the following equivalent conditions are satisfied:

  1. When endowed with its canonical uniformity, X becomes is a complete uniform space.
  2. X is a complete subset of itself.
  3. There exists a neighborhood of the origin in (X, τ) that is also a complete subset of X.[5]
    • This implies that every locally compact TVS is complete (even if the TVS is not Hausdorff).
  4. Every Cauchy prefilter 𝒞 ⊆ ℘(X) on X converges in (X, τ) to at least one point of X.
    • If X is Hausdorff then every prefilter on X will converge to at most one point of X. But if X is not Hausdorff then a prefilter may converge to multiple points in X. The same is true for nets.
  5. Every Cauchy filter on X converges in (X, τ) to at least one point of X.
  6. Every Cauchy net in X converges in (X, τ) to at least one point of X.

where if in addition X is pseudometrizable or metrizable (e.g. a normed space) then this list can be extended to include:

  1. (X, τ) is sequentially complete.

A topological vector space (X, τ) is sequentially complete if any of the following equivalent conditions are satisfied:

  1. X is a sequentially complete subset of itself.
  2. Every Cauchy sequence in X converges in (X, τ) to at least one point of X.
  3. Every elementary Cauchy prefilter on X converges in (X, τ) to at least one point of X.
  4. Every elementary Cauchy filter on X converges in (X, τ) to at least one point of X.

Uniqueness of the canonical uniformity
See also: Topological group

The existence of the canonical uniformity was demonstrated above by defining it. The theorem below establishes that the canonical uniformity of any TVS (X, τ) is the only uniformity on X that is both (1) translation invariant, and (2) generates on X the topology τ.

Theorem[7] (Existence and uniqueness of the canonical uniformity) — The topology of any TVS can be derived from a unique translation-invariant uniformity. If 𝒩(0) is any neighborhood base of the origin, then the family { Δ(N)  :  N ∈ 𝒩(0) } is a base for this uniformity.

This section is dedicated to explaining the precise meanings of the terms involved in this uniqueness statement.

Uniform spaces and translation-invariant uniformities
Main article: Uniform space

Let S ⊆ X, let Φ and Ψ be subsets of X × X, and let p ∈ X. Define[1]

Φop      ≝  { (y, x)  :  (x, y) ∈ Φ }
Φ ∘ Ψ  ≝  { (x, z)  :  ∃ y ∈ X  such that  (x, y) ∈ Ψ and (y, z) ∈ Φ }  =  ∪y ∈ X { (x, z)  :  (x, y) ∈ Ψ and (y, z) ∈ Φ }
S ⋅ Φ  ≝  { y ∈ X : Φ ∩ (S × { y }) ≠ ∅ }  =  Pr2(Φ ∩ (S × X))
Φ ⋅ S  ≝  { x ∈ X : Φ ∩ ({ x } × S) ≠ ∅ }  =  Pr1(Φ ∩ (X × S))  =  S ⋅ Φop
p ⋅ Φ  ≝  { p } ⋅ Φ = { y ∈ X  :  (p, y) ∈ Φ }
Φ ⋅ p  ≝  Φ ⋅ { p } = { x ∈ X  :  (x, p) ∈ Φ }  =  p ⋅ Φop

where Φ ⋅ S (resp. S ⋅ Φ) is called the set of left (resp. right) Φ-relatives of (points in) S. The maps Pr1 and Pr2 are the canonical projections onto the first and second coordinates, respectively. Given Φ ⊆ X × X, Φ is symmetric if Φ = Φop while a subset S ⊆ X is called Φ-small if S × S ⊆ Φ. Two points x and y are Φ-close if (x, y) ∈ Φ.

If Φ, Ψ, Ω ⊆ X × X and R, S ⊆ X then:

  • Φ ⊆ Ψ if and only if Φ op ⊆ Ψ op. Moreover, (Φ ∘ Ψ)op  =  Ψ op ∘ Φ op and Φ ∩ Ψ is symmetric.
  • If Φ ⊆ Φ2 and Ψ ⊆ Ψ2 then Φ ∘ Ψ  ⊆  Φ2 ∘ Ψ2.
  • Associativity: Φ ∘ (Ψ ∘ Ω)  =  (Φ ∘ Ψ) ∘ Ω.
  • (Φ ∘ Ψ) ⋅ S = Φ ⋅ (Ψ ⋅ S).

A non-empty family ℬ ⊆ ℘(X × X) is called a base or fundamental system of entourages if ℬ is a prefilter on X × X satisfying all of the following conditions:

  1. Every set in ℬ contains the diagonal of X as a subset; that is, ΔX  ≝  { (x, x) : x ∈ X } ⊆ Φ for every Φ ∈ ℬ. Said differently, the prefilter ℬ is fixed on ΔX.
  2. For every Ω ∈ ℬ there exists some Φ ∈ ℬ such that Φ ∘ Φ ⊆ Ω.
  3. For every Ω ∈ ℬ there exists some Φ ∈ ℬ such that Φ ⊆ Ωop.

A uniformity or uniform structure on X is a filter 𝒰 on X × X that is generated by some base of entourages ℬ, in which case we say that ℬ is a base of entourages for 𝒰.

For a commutative additive group X, a fundamental system of entourages ℬ is called translation-invariant[7] if for every Φ ∈ ℬ, (x, y) ∈ Φ if and only if (x + z, y + z) ∈ Φ for all x, y, z ∈ X. A uniformity ℬ is called translation-invariant[7] if it has a base of entourages that is translation-invariant. The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin. The canonical uniformity on any TVS is translation-invariant.[7]

Topology generated by a uniformity
See also: Topological space § Neighborhood definition, and Characterizations of the category of topological spaces

Let ℬ ⊆ ℘(X × X) be a base of entourages on X. For every S ⊆ X and p ∈ X, the neighborhood prefilter on S (resp. at p) is the set

ℬ ⋅ S  ≝  { Φ ⋅ S  :  Φ ∈ ℬ }       and       ℬ ⋅ p  ≝  ℬ ⋅ { p }  =  { Φ ⋅ p  :  Φ ∈ ℬ }

and the filter on X that it generates is known as the neighborhood filter of S (resp. of p). The assignment

x ∈ X ↦ ℬ ⋅ x ≝ { Φ ⋅ x  :  Φ ∈ ℬ }

of points to prefilters generates a topology on X called the topology induced by ℬ. A subset U ⊆ X is open in this topology if and only if any of the following equivalent conditions hold:

  1. For every u ∈ U there exists some N ∈ ℬ ⋅ u such that N ⊆ U.
  2. For every u ∈ U there exists some Φ ∈ ℬ such that Φ ⋅ u = { x ∈ X  :  (x, u) ∈ Φ } ⊆ U.

The closure of a subset S ⊆ X in this topology is:

 ClX S  =  ∩Φ ∈ ℬ (Φ ⋅ S)  =  ∩Φ ∈ ℬ (S ⋅ Φ).
Cauchy prefilters and complete uniformities

A prefilter ℱ ⊆ ℘(X) on a uniform space X with uniformity 𝒰 is called a Cauchy prefilter if for every entourages N ∈ 𝒰, there exists some F ∈ ℱ such that F × F ⊆ N.

A uniform space (X, 𝒰) is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on X converges to at least one point of X when X is endowed with the topology induced by 𝒰.

If (X, τ) is a TVS then for any S ⊆ X and x ∈ X,

ΔX(N) ⋅ S  =  S + N      and      ΔX(N) ⋅ x  =  x + N.

The topology induced on X by the canonical uniformity is the same as the topology that X started with (i.e. it is τ).

TVS completeness vs completeness of (pseudo)metrics

Preliminaries: Complete pseudometric spaces
Main articles: Complete metric space, Pseudometric space, and Cauchy sequence

We review the basic notions related to the general theory of complete pseudometric spaces. Recall that every metric is a pseudometric and that a pseudometric p is a metric if and only if p(x, y) = 0 implies x = y. Thus every metric space is a pseudometric space and a pseudometric space (X, p) is a metric space if and only if p is a metric.

If S is a subset of a pseudometric space (X, d) then the diameter of S is defined to be

diam(S) ≝ sup { d (s, t) : s, t ∈ S  }.

A prefilter ℬ on a pseudometric space (X, d) is called a d-Cauchy prefilter or simply a Cauchy prefilter if for each real r > 0, there is some B ∈ ℬ such that the diameter of B is less than r.

Suppose (X, d) is a pseudometric space. A net x• = (xi)i ∈ I in X is called a d-Cauchy net or simply a Cauchy net if Tails (x•) is a Cauchy prefilter, which happens if and only if

for every r > 0 there is some i ∈ I such that if j, k ∈ I with j  ≥  i and k  ≥  i then d (xj, xjn)  <  r.

A Cauchy sequence is a sequence that is also a Cauchy net.[note 2]

Every pseudometric p on a set X induces the usual canonical topology on X, which we'll denote by τp; it also induces a canonical uniformity on X, which we'll denote by 𝒰p. The topology on X induced by the uniformity 𝒰p is equal to τp. A net x• = (xi)i ∈ I in X is Cauchy with respect to p if and only if it is Cauchy with respect to the uniformity 𝒰p. The pseudometric space (X, p) is a complete (resp. a sequentially complete) pseudometric space if and only if (X, 𝒰p) is a complete (resp. a sequentially complete) uniform space. Moreover, the pseudometric space (X, p) (resp. the uniform space (X, 𝒰p)) is complete if and only if it is sequentially complete.

A pseudometric space (X, d) (e.g. a metric space) is called complete and d is called a complete pseudometric if any of the following equivalent conditions hold:

  1. Every Cauchy prefilter on X converges to at least one point of X.
  2. The above statement but with the word "prefilter" replaced by "filter."
  3. Every Cauchy net in X converges to at least one point of X.
  4. Every Cauchy sequence in X converges to at least one point of X.
    • Thus to prove that (X, d) is complete, it suffices to only consider Cauchy sequences in X (and it is not necessary to consider the more general Cauchy nets).
  5. The canonical uniformity on X induced by the pseudometric d is a complete uniformity.
  • Note that if d is a metric on X then any such limit point is necessarily unique and the same is true for limits of Cauchy prefilters on X.

And if addition d is a metric then we may add to this list:

  1. Every decreasing sequence of closed balls whose diameters shrink to 0 has non-empty intersection.[8]

Complete pseudometrics and complete TVSs

Every F-space, and thus also every Fréchet space, Banach space, and Hilbert space is a complete TVS. Note that every F-space is a Baire space but there are normed spaces that are Baire but not Banach.[9]

A pseudometric d on a vector space X is said to be translation invariant if d(x,  y) = d(x + z,  y + z) for all vectors x, y, z ∈ X.

Suppose (X, τ) is pseudometrizable TVS (e.g. a metrizable TVS) and that p is any pseudometric on X such that the topology on X induced by p is equal to τ. If p is translation-invariant, then (X, τ) is a complete TVS if and only if (X, p) is a complete pseudometric space.[10] If p is not translation-invariant, then may be possible for (X, τ) to be a complete TVS but (X, p) to not be a complete pseudometric space[10] (see this footnote[note 3] for an example).[10]

Theorem[11][12] (Klee) — Let d be any[note 4] metric on a vector space X such that the topology 𝜏 induced by d on X makes (X, 𝜏) into a topological vector space. If (X, d) is a complete metric space then (X, 𝜏) is a complete-TVS.

Complete norms and equivalent norms

Two norms on a vector space are called equivalent if and only if they induce the same topology.[13] If p and q are two equivalent norms on a vector space X then the normed space (X, p) is a Banach space if and only if (X, q) is a Banach space. See this footnote for an example of a continuous norm on a Banach space that is not is not equivalent to that Banach space's given norm.[note 5][13] All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.[14]

Completions

A completion[15] of a TVS X is a complete TVS that contains a dense vector subspace that is TVS-isomorphic to X.

Non-uniqueness of all completions

As the example below shows, regardless of whether or not a space is Hausdorff or already complete, every topological vector space (TVS) has infinitely many non-isomorphic completions.[16]

However, every Hausdorff TVS has a Hausdorff completion that is unique up to TVS-isomorphism.[16] But nevertheless, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions.

Example (Non-uniqueness of completions):[15] Let C denote any complete TVS and let I denote any TVS endowed with the indiscrete topology, which recall makes I into a complete TVS. Since both I and C are complete TVSs, so is their product I × C. If U and V are non-empty open subsets of I and C, respectively, then U = I and (U × V) ∩ ({ 0 } × C) = { 0 } × V ≠ ∅, which shows that { 0 } × C is a dense subspace of I × C. Thus by definition of "completion," I × C is a completion of { 0 } × C  (it doesn't matter that { 0 } × C is already complete). So by identifying { 0 } × C with C, if X ⊆ C is a dense vector subspace of C, then X has both C and I × C as completions.

Hausdorff completions

Every Hausdorff TVS has a Hausdorff completion that is unique up to TVS-isomorphism.[16] But nevertheless, as shown above, every Hausdorff TVS still has infinitely many non-isomorphic non-Hausdorff completions.

Properties of Hausdorff completions[17] — Suppose that X and C are Hausdorff TVSs with C complete. Suppose that E : X → C is a TVS-embedding onto a dense vector subspace of C. Then

Universal property: for every continuous linear map f : X → Z into a complete Hausdorff TVS Z, there exists a unique continuous linear map F : C → Z such that f = F ∘ E.

If E2 : X → C2 is a TVS embedding onto a dense vector subspace of a complete Hausdorff TVS C2 having the above universal property, then there exists a unique (bijective) TVS-isomorphism I : C → C2 such that E2 = I ∘ E.

Corollary[17] — Suppose C is a complete Hausdorff TVS and X is a dense vector subspace of C. Then every continuous linear map f : X → Z into a complete Hausdorff TVS Z has a unique continuous linear extension to a map C → Z.

Existence of Hausdorff completions
See also: Filters in topology

A Cauchy filter ℬ on a TVS X is called a minimal Cauchy filter[17] if there does not exist a Cauchy filter on X that is strictly coarser than ℬ (i.e. "strictly coarser than ℬ" means contained as a proper subset of ℬ).

If ℬ is a Cauchy filter on X then the filter generated by the following prefilter:

{ B + N  :  B ∈ ℬ and N is a neighborhood of 0 in X }

is the unique minimal Cauchy filter on X that is contained as a subset of ℬ.[17] In particular, for any x ∈ X, the neighborhood filter at x is a minimal Cauchy filter.

Let 𝕄 be the set of all minimal Cauchy filters on X and let E : X → 𝕄 be the map defined by sending x ∈ X to the neighborhood filter of x in X. We endow 𝕄 with the following vector space structure. Given ℬ, 𝒞 ∈ 𝕄 and a scalar s, let ℬ + 𝒞 (resp. s ℬ) denote the unique minimal Cauchy filter contained in the filter generated by { B + C  :  B ∈ ℬ and C ∈ 𝒞 } (resp. { sB  :  B ∈ ℬ  }).

For every balanced neighborhood N of 0 in X, let

𝕌(N)  ≝  { ℬ ∈ 𝕄  :  ∃ B ∈ ℬ and ∃ a neighborhood V of 0 in X such that B + V ⊆ N  }.

If X is Hausdorff then the collection of all sets 𝕌(N), as N ranges over all balanced neighborhoods of 0 in X, forms a vector topology on 𝕄 making 𝕄 into a complete Hausdorff TVS. Moreover, the map E : X → 𝕄 is a TVS-embedding onto a dense vector subspace of 𝕄.[17]

If X is a metrizable TVS then a Hausdorff completion of X can be constructed using equivalence classes of Cauchy sequences instead of minimal Cauchy filters.

Non-Hausdorff completions

We now show how every non-Hausdorff TVS X can be TVS-embedded onto a dense vector subspace of a complete TVS. The proof that every Hausdorff TVS has a Hausdorff completion is widely available so we use its conclusion to prove that every non-Hausdorff TVS also has a completion. These details are sometimes useful for extending results from Hausdorff TVSs to non-Hausdorff TVSs.

Let I = cl { 0 } denote the closure of the origin in X, where I is endowed with its subspace topology induced by X (so that I has the indiscrete topology). Since I has the trivial topology, it is easily shown that every vector subspace of X that is an algebraic complement of I in X is necessarily a topological complement of I in X.[18][19] Let H denote any topological complement of I in X, which is necessarily a Hausdorff TVS (since it is TVS-isomorphic to the quotient TVS X / I [note 6]). Since X is the topological direct sum of I and H (which means that X = I ⊕ H in the category of TVSs), the canonical map

I × H  →  I ⊕ H = X,     given by     (x, y)  ↦  x + y

is a TVS-isomorphism.[19] Let A  :  X = I ⊕ H  →  I × H denote the inverse of this canonical map. (As a side note, it follows that every open and every closed subset U of X satisfies U = I + U.[proof 1])

The Hausdorff TVS H can be TVS-embedded, say via the map InH  :  H  →  C, onto a dense vector subspace of its completion C. Since I and C are complete, so is their product I × C. Let IdI : I → I denote the identity map and observe that the product map IdI × InH  :  I × H  →  I × C is a TVS-embedding whose image is dense in I × C. Define the map[note 7]

B  :  X = I ⊕ H  →  I × C     by     B  ≝  (IdI × In)  ∘  A

which is a TVS-embedding of X = I ⊕ H onto a dense vector subspace of the complete TVS I × C. Moreover, observe that the closure of the origin in I × C is equal to I × { 0  }, and that I × { 0 } and { 0 } × C are topological complements in I × C.

To summarize,[19] given any algebraic (and thus topological) complement H of I ≝ cl { 0 } in X and given any completion C of the Hausdorff TVS H such that H ⊆ C, then the natural inclusion[20]

InH  :  X = I ⊕ H  →  I ⊕ C

is a well-defined TVS-embedding of X onto a dense vector subspace of the complete TVS  I ⊕ C where moreover, we have

X  =  I ⊕ H  ⊆  I ⊕ C  ≅  I × C.

Topology of a completion

Theorem[7][21] (Topology of a completion) — Let C be a complete TVS and let X be a dense vector subspace of X. If 𝒩X(0) is any neighborhood base of the origin in X then the set

𝒩C(0)  ≝  { clC N  :  N ∈ 𝒩X(0) }

is a neighborhood of the origin in the completion C of X.

If X is locally convex and 𝒫 is a family of continuous seminorms on X that generate the topology of X, then the family of all continuous extensions to C of all members of 𝒫 is a generating family of seminorms for X.

Said differently, if C is a completion of a TVS X with X ⊆ C and if 𝒩 is a neighborhood base of the origin in X, then the family of sets

{ ClC N  :  N ∈ 𝒩 }

is a neighborhood basis at the origin in C.[3]

Theorem[22] (Completions of quotients) — Let M be a metrizable topological vector space and let N be a closed vector subspace of M. Suppose that C is a completion of M. Then the completion of M/N is TVS-isomorphic to C / clC N. If in addition M is a normed space, then this TVS-isomorphism is also an isometry.

Properties preserved by completions

If a TVS X has any of the following properties then so does its completion:

Other preserved properties

If X is a Hausdorff TVS, then the continuous dual space of X is identical to the continuous dual space of the completion of X.[27] The completion of a locally convex bornological space is a barrelled space.[24] If X and Y are DF-spaces then the projective tensor product, as well as its completion, of these spaces is a DF-space.[28]

The completion of the projective tensor product of two nuclear spaces is nuclear.[22] The completion of a nuclear space is TVS-isomorphic with a projective limit of Hilbert spaces.[22]

Properties of maps preserved by extensions to a completion

If f : X → Y is a nuclear linear operator between two locally convex spaces and if C be a completion of X then f has a unique continuous linear extension to a nuclear linear operator F : C → Y.[22]

Let X and Y be two Hausdorff TVSs with Y complete. Let C be a completion of X. Let L(X; Y) denote the vector space of continuous linear operators and let I : L(X; Y) → L(C; Y) denote the map that sends every f ∈ L(X; Y) to its unique continuous linear extension on C. Then I : L(X; Y) → L(C; Y) is a (surjective) vector space isomorphism. Moreover, I : L(X; Y) → L(C; Y) maps families of equicontinuous subsets onto each other. Suppose that L(X; Y) is endowed with a 𝒢-topology and that ℋ denotes the closures in C of sets in 𝒢. Then the map I : L𝒢(X; Y) → Lℋ(C; Y) is also a TVS-isomorphism.[22]

Examples and sufficient conditions for a complete TVS

Theorem — [11] Let d be any (not assumed to be translation-invariant) metric on a vector space X such that the topology 𝜏 induced by d on X makes (X, 𝜏) into a topological vector space. If (X, d) is a complete metric space then (X, 𝜏) is a complete-TVS.

  • Any TVS endowed with the trivial topology is complete and every one of its subsets is complete. Moreover, every TVS with the trivial topology is compact and hence locally compact. Thus a complete seminormable locally convex and locally compact TVS need not be finite-dimensional if it is not Hausdorff.
  • An arbitrary product of complete (resp. sequentially complete, quasi-complete) TVSs has that same property. If all spaces are Hausdorff, then the converses are also true.[29] A product of Hausdorff completions of a family of (Hausdorff) TVSs is a Hausdorff completion of their product TVS.[29] More generally, an arbitrary product of complete subsets of a family of TVSs is a complete subset of the product TVS.[30]
  • The projective limit of a projective system of Hausdorff complete (resp. sequentially complete, quasi-complete) TVSs has that same property.[29] A projective limit of Hausdorff completions of an inverse system of (Hausdorff) TVSs is a Hausdorff completion of their projective limit.[29]
  • If M is a closed vector subspace of a complete pseudometrizable TVS X, then the quotient space X ∖ M is complete.[3]
  • Suppose M is a complete vector subspace of a metrizable TVS X. If the quotient space X ∖ M is complete then so is X.[3][31] However, there exists a complete TVS X having a closed vector subspace M such that the quotient TVS X / M is not complete.[17]
  • Every F-space, Fréchet space, Banach space, and Hilbert space is a complete TVS.
  • Strict LF-spaces and strict LB-spaces are complete.[32]
  • Suppose that D is a dense subset of a TVS X. If every Cauchy filter on D converges to some point in X then X is complete.[31]
  • The Schwartz space of smooth functions is complete.
  • The spaces of distributions and test functions is complete.
  • Suppose that X and Y are locally convex TVSs and that the space of continuous linear maps Lb(X; Y) is endowed with the topology of uniform convergence on bounded subsets of X. If X is a bornological space and if Y is complete then Lb(X; Y) is a complete TVS.[32] In particular, the strong dual of a bornological space is complete.[32] However, it need not be bornological.
  • Every quasi-complete DF-space is complete.[26]
  • Let ω and τ be Hausdorff TVS topologies on a vector space X such that ω ⊆ τ. If there exists a prefilter ℬ such that ℬ is a neighborhood basis at 0 for (X, τ) and such that every B ∈ ℬ is a complete subset of (X, ω), then (X, τ) is a complete TVS.[5]

Properties

Complete TVSs

Every TVS has a completion and every Hausdorff TVS has a Hausdorff completion.[33] Every complete TVS is quasi-complete space and sequentially complete.[34] However, the converses of the above implications are generally false.[34] There exists a sequentially complete locally convex TVS that is not quasi-complete.[26]

If a TVS has a complete neighborhood of the origin then it is complete.[35] Every complete pseudometrizable TVS is a barrelled space and a Baire space (and thus non-meager).[36] The dimension of a complete metrizable TVS is either finite or uncountable.[19]

Cauchy nets and prefilters

Any neighborhood basis of any point in a TVS is a Cauchy prefilter.

Every convergent net (resp. prefilter) in a TVS is necessarily a Cauchy net (resp. a Cauchy prefilter).[5] Any prefilter that is subordinate to (i.e. finer than) a Cauchy prefilter is necessarily also a Cauchy prefilter[5] and any prefilter finer than a Cauchy prefilter is also a Cauchy prefilter. The filter associated with a sequence in a TVS is Cauchy if and only if the sequence is a Cauchy sequence. Every convergent prefilter is a Cauchy prefilter.

If X is a TVS and if x ∈ X is a cluster point of a Cauchy net (resp. Cauchy prefilter), then that Cauchy net (resp. that Cauchy prefilter) converges to x in X.[3] If a Cauchy filter in a TVS has an accumulation point x then it converges to x.

Uniformly continuous maps send Cauchy nets to Cauchy nets.[3] A Cauchy sequence in a Hausdorff TVS X is not necessarily relatively compact (i.e. its closure in X is not necessarily compact).

Suppose that X• = (Xi)i ∈ I is a family of TVSs and that X denotes the product of these TVSs. Suppose that for every index i, ℬi is a prefilter on Xi. Then the product of this family of prefilters is a Cauchy filter on X if and only if each ℬi is a Cauchy filter on Xi.[17]

Maps

If f : X → Y is an injective topological homomorphism from a complete TVS into a Hausdorff TVS then the range of f (i.e. f (X)) is a closed subspace of Y.[31] If f : X → Y is a topological homomorphism from a complete metrizable TVS into a Hausdorff TVS then the range of f is a closed subspace of Y.[31] If f : X → Y is a uniformly continuous map between two Hausdorff TVSs then the image under f of a totally bounded subset of X is a totally bounded subset of Y.[37]


Uniformly continuous extensions

Suppose that f : D → Y is a uniformly continuous map from a dense subset D of a TVS X into a complete Hausdorff TVS Y. Then f has a unique uniformly continuous extension to all of X.[3] If in addition f is a homomorphism then its unique uniformly continuous extension is also a homomorphism.[3] This remains true if "TVS" is replaced by "commutative topological group."[3] The map f is not required to be a linear map and that D is not required to be a vector subspace of X.

Uniformly continuous linear extensions

Suppose f : X → Y be a continuous linear operator between two Hausdorff TVSs. If M is a dense vector subspace of X and if the restriction f |M : M → Y to M is a topological homomorphism then f : X → Y is also a topological homomorphism.[38] So if C and D are Hausdorff completions of X and Y, respectively, and if f : X → Y is a topological homomorphism, then f 's unique continuous linear extension F : C → D is a topological homomorphism. (Note that it's possible for f : X → Y to be surjective but for F : C → D to not be injective.)[38]

Suppose X and Y are Hausdorff TVSs, M is a dense vector subspace of X, and N is a dense vector subspaces of Y. If M are and N are topologically isomorphic additive subgroups via a topological homomorphism f then the same is true of X and Y via the unique uniformly continuous extension of f (which is also a homeomorphism).[39]

Subsets
Complete subsets

Every complete subset of a TVS is sequentially complete. A complete subset of a Hausdorff TVS X is a closed subset of X.[3][35]

Every compact subset of a TVS is complete (even if the TVS is not Hausdorff or not complete).[3][35] Closed subsets of a complete TVS are complete; however, if a TVS X is not complete then X is a closed subset of X that is not complete. The empty set is complete subset of every TVS. If C is a complete subset of a TVS (the TVS is not necessarily Hausdorff or complete) then any subset of C that is closed in C is complete.[35]

Topological complements

If X is a non-normable Fréchet space on which there exists a continuous norm then X contains a closed vector subspace that has no topological complement.[26] If X is a complete TVS and M is a closed vector subspace of X such that X / M is not complete, then H does not have a topological complement in X.[26]

Subsets of completions

Let M be a separable locally convex metrizable topological vector space and let C be its completion. If S is a bounded subset of C then there exists a bounded subset R of X such that S ⊆ clC R.[26]

Relation to compact subsets

A subset of a TVS (not assumed to be Hausdorff or complete) is compact if and only if it is complete and totally bounded.[40][proof 2] Thus a closed and totally bounded subset of a complete TVS is compact.[41][3]

Every complete totally bounded set is relatively compact.[3] If X is any TVS then the quotient map q : X → X / clX({ 0 }) is a closed map[42] and thus S + clX ({ 0 })  ⊆  ClX S A subset S of a TVS X is totally bounded if and only if its image under the canonical quotient map q : X → X / clX({ 0 }) is totally bounded.[19] Thus S is totally bounded if and only if S + clX ({ 0 }) is totally bounded. In any TVS, the closure of a totally bounded subset is again totally bounded.[3] In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.[33] If S is a subset of a TVS X such that every sequence in S has a cluster point in S then S is totally bounded.[19] A subset S of a Hausdorff TVS X is totally bounded if and only if every ultrafilter on S is Cauchy, which happens if and only if it is pre-compact (i.e. its closure in the completion of X is compact).[37]

If S ⊆ X is compact, then clX S  = S + clX { 0 } and this set is compact. Thus the closure of a compact set is compact[note 8] (i.e. all compact sets are relatively compact).[43] Thus the closure of a compact set is compact. Every relatively compact subset of a Hausdorff TVS is totally bounded.[37]

In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.[33] More generally, if K is a compact subset of a locally convex space, then the convex hull co K (resp. the disked hull cobal K) is compact if and only if it is complete.[33] Every subset S of cl({ 0 }) is compact and thus complete.[proof 3] In particular, if X is not Hausdorff then there exist compact complete sets that are not closed.[3]

See also

Notes
  1. Completeness of normed spaces and metrizable TVSs are defined in terms of norms and metrics. In contrast to the uniqueness of this translation-invariant uniformity, in general, many different norms (e.g. equivalent norms) and metrics may be used to determine completeness of such space.
  2. Every sequence is also a net.
  3. The normed space (ℝ, |⋅|) is a Banach space where the absolute value is a norm that induces the usual Euclidean topology on ℝ. Define a metric D on {{math|ℝ} by D(x, y) = |arctan (x) - arctan (y)| for all x, y ∈ ℝ, where one may show that D induces the usual Euclidean topology on ℝ. However, D is not a complete metric since the sequence x• = (xi)∞
    i=1     defined by xi = i is a D-Cauchy sequence that does not converge in ℝ to any point of ℝ. Note also that this D-Cauchy sequence is not a Cauchy sequence in (ℝ, |⋅|) (i.e. it is not a Cauchy sequence with respect to the norm |⋅|).
  4. Not assumed to be translation-invariant.
  5. Let (C([0, 1]), || ⋅ ||∞) denotes the Banach space of continuous functions with the supremum norm, let X = C([0, 1]) where X is given the topology induced by || ⋅ ||∞), and denote the restriction of the L1-norm to C([0, 1]) by || ⋅ ||1. Then one may show that || ⋅ ||1 ≤ || ⋅ ||∞ so that the norm || ⋅ ||1 : X → ℝ is a continuous function. However, || ⋅ ||1 is not equivalent to the norm || ⋅ ||∞) and so in particular, (C([0, 1]), || ⋅ ||1) is not a Banach space.
  6. This particular quotient map q : X → X / I is in fact also a closed map.
  7. Explicitly, this map is defined as follows: for each x ∈ X, let (i, h) = A (x) and so that B(x) ≝ (i, InH h). Then B (i + h) = (i, InH h) holds for all i ∈ I and h ∈ H.
  8. In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (e.g. the particular point topology on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. The proof uses and that fact that S is compact (but possibly not closed) and clX { 0 } is both closed and compact so that S + clX { 0 } , which is the image of the compact set S × clX { 0 } under the continuous addition map ⋅ + ⋅  :  X × X → X, is also compact. Recall also that the sum of a compact set (i.e. S) and a closed set is closed so S + clX { 0 } is closed in X.
Proofs
  1. Let W be a neighborhood of 0 in X. Since A(W) is a neighborhood of 0 in I × H we can pick an open (resp. closed) neighborhood V of 0 in H such that I × V ⊆ A(W) is a neighborhood of the origin. Clearly, V is open (resp. closed) if and only if I × V is open (resp. closed). Let U = I + V so that A(U) = I × V ⊆ A(W) where U is open (resp. closed) if and only if V is open (resp. closed).
  2. Suppose S is compact in X and let 𝒞 be a Cauchy filter on S. Let 𝒟 = { clS C : C ∈ 𝒞 so that 𝒟 is a Cauchy filter of closed sets. Since 𝒟 has the finite intersection property, there exists some s ∈ S such that s clS C for all C ∈ 𝒞 so s ∈ cl 𝒞 (i.e. s is an accumulation point of 𝒞). Since 𝒞 is Cauchy, 𝒞 → x in S. Thus S is complete. That S is also totally bounded follows immediately from the compactness of S.
  3. Given any open cover of S, pick any open set U from that cover that contains the origin. Since U is a neighborhood of the origin, U contains cl({ 0 }) and thus contains S.

Citations
  1. Schaefer & Wolff 1999, pp. 1-11.
  2. Edwards 1995, p. 61.
  3. Narici & Beckenstein 2011, pp. 47-66.
  4. Narici & Beckenstein 2011, p. 48.
  5. Narici & Beckenstein 2011, pp. 48-51.
  6. Narici & Beckenstein 2011, pp. 48–51.
  7. Schaefer & Wolff 1999, pp. 12-19.
  8. Narici & Beckenstein 2011, pp. 64-66.
  9. Wilansky 2013, p. 29.
  10. Narici & Beckenstein 2011, pp. 47-51.
  11. Schaefer & Wolff 1999, p. 35.
  12. Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
  13. Conrad, Keith. "Equivalence of norms" (PDF). kconrad.math.uconn.edu. Retrieved September 7, 2020.
  14. see Corollary 1.4.18, p. 32 in Megginson (1998).
  15. Narici & Beckenstein 2011, pp. 60-61.
  16. Narici & Beckenstein 2011, pp. 93-113.
  17. Horváth 1966, pp. 139-141.
  18. Wilansky 2013, p. 63.
  19. Schaefer & Wolff 1999, pp. 12-35.
  20. where for all i ∈ I and h ∈ H, InH(i + h)  ≝  i + h.
  21. Schaefer & Wolff 1999, pp. 36-72.
  22. Schaefer & Wolff 1999, pp. 73-121.
  23. Trèves 2006, pp. 112-125.
  24. Schaefer & Wolff 1999, pp. 68-72.
  25. Schaefer & Wolff 1999, pp. 122-202.
  26. Schaefer & Wolff 1999, pp. 190-202.
  27. Narici & Beckenstein 2011, pp. 225–273.
  28. Schaefer & Wolff 1999, pp. 199-202.
  29. Jarchow 1981, pp. 56-73.
  30. Narici & Beckenstein 2011, p. 57.
  31. Horváth 1966, pp. 129-141.
  32. Narici & Beckenstein 2011, pp. 441-457.
  33. Narici & Beckenstein 2011, pp. 67-113.
  34. Narici & Beckenstein 2011, pp. 155-176.
  35. Narici & Beckenstein 2011, pp. 115-154.
  36. Narici & Beckenstein 2011, pp. 371-423.
  37. Horváth 1966, pp. 145-149.
  38. Schaefer & Wolff 1999, p. 116.
  39. Narici & Beckenstein 2011, p. 59.
  40. Narici & Beckenstein 2011, pp. 55-56.
  41. Narici & Beckenstein 2011, pp. 55-66.
  42. Narici & Beckenstein 2011, pp. 107-112.
  43. Narici & Beckenstein 2011, p. 156.

Bibliography
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