Complete topological vector space

In functional analysis and related areas of mathematics, a topological vector spaces (TVS) is complete if its canonical uniformity is complete.

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. All topological vector spaces, even those that are not metrizable or Haussddorff, have a completion.

Definitions and notation
Main article: Filter (mathematics)

Throughout, X will be a non-empty set and 𝒜 and ℬ will be collections of subsets of X.

The theory of filters and filter bases is well developed and has many definitions and notations, many of which we now unceremoniously list to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. We describe many of their important properties later. Note that not all notation related to filters is well established and some notation varies greatly across the literature (e.g. the notation for the set of all filter bases on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

NotationDefinitionName
℘(X)  :=  { S : S ⊆ X } Set of all subsets of X Power set of a set X[1]
Top(X) Set of all topologies on X
Filters(X) Set of all filters on X
PreFilters(X) = FilterBases(X) Set of all filter bases on X
Func(X; Y) Set of all functions from X into Y
Sets operations
Definition:[2] The upward closure or isotonization of a collection ℬ of subsets of X is
ℬ↑  :=  ℬ↑X  :=  { S ⊆ X : B ⊆ S for some B ∈ ℬ }  =  ∪B ∈ ℬ { S : B ⊆ S ⊆ X }.
Notation and DefinitionAssumptionsName
ker ℬ  :=  ∩B ∈ ℬ B Kernel[1]
ℬ ∩ { S }  :=  { B ∩ S : B ∈ ℬ } S ⊆ X Trace of ℬ on S[3]
S ∖ ℬ  :=  { S ∖ B : B ∈ ℬ } S ⊆ X Set subtraction[3]
ℬ↓  :=  ℬ↓X  =  ∪B ∈ ℬ { S : S ⊆ B } S ⊆ X Downward closure[1]
𝒜 + ℬ  :=  { A + B : A ∈ 𝒜, B ∈ ℬ } X is an additive group Sum[3]
s ℬ  :=  { s B : B ∈ ℬ } X is a vector space and s is a scalar Scalar multiple[3]
f (ℬ)  :=  { f (B) : B ∈ ℬ } f : X → Y is a map Image of ℬ under f[4]
f -1 (𝒞)  :=  { f -1 (C) : C ∈ 𝒞 } f : X → Y is a map and 𝒞 ⊆ ℘(Y) Preimage of ℬ under f[4]
Im f  :=  f (X)  =  { f (x) : x ∈ X } f : X → Y is a map Image or range of f
S↑X  :=  { S }↑X S ⊆ X Upward closure/Isotonization[1]
𝒜 ⊓ ℬ  :=  𝒜 ∩ ℬ  :=  { A ∩ B : A ∈ 𝒜, B ∈ ℬ } Pairwise intersection[3]

Nets and topologies
Directed sets and nets notation.
Definition:[5] A directed set is a set I together with a preorder, which we will assume is denoted by ≤ (unless otherwise specified), that makes (I, ≤) into an upward directed set, which means that for every i, j ∈ I, there exists some k ∈ I such that i ≤ k and j ≤ k. We define j ≥ i to mean i ≤ j. A net in X is a map from a directed set into X.
Notation and DefinitionAssumptionsName
I≥ i  :=  { j ∈ I : i ≤ j } i ∈ I and (I, ≤) is a directed set Tail of I starting at i
I> i  :=  { j ∈ I : i ≤ j,  j ≠ i } i ∈ I and (I, ≤) is a directed set Tail of I after i
f (I≥ i)  =  { f (j) : i ≤ j,  j ∈ I } i ∈ I and f : (I, ≤) → X is a net Tail of f starting at i[6]
f (I> i)  =  { f (j) : i ≤ j,  j ≠ i,  j ∈ I } i ∈ I and f : (I, ≤) → X is a net Tail of f after i
(x•)i  :=  x≥ i  :=  { xj : i ≤ j,  j ∈ I } i ∈ I and x• = (xi)i ∈ I is a net Tail of x• starting at i
x> i  :=  { xj : i ≤ j,  j ≠ i,  j ∈ I } i ∈ I and x• = (xi)i ∈ I is a net Tail of x• after i
Tails(x•)  :=  (xi)
i  :=  x≥ •  :=  { x≥ i : i ∈ I }		 x• = (xi)i ∈ I is a net Set/(Eventuality) filter base of/associated with/generated by (tails of) x•. If x• is a sequence it is called the sequential filter base instead.[6]
TailsFilter(x•)  :=  Tails(x•)↑X x• = (xi)i ∈ I is a net (Eventuality) filter of/associated with/generated by (tails of) x•[6]


Topology notation

If τ is a topology on X then we may use the following notation.

Notation and DefinitionAssumptionsName
τ (S)  :=  { O ∈ τ : S ⊆ O } S ⊆ X (Filter base of) Open neighborhoods of S in (X, τ)
τ (x)  :=  { O ∈ τ : x ∈ O } x ∈ X (Filter base of) Open neighborhoods of x in (X, τ)
𝒩τ(S)  :=  𝒩(S)  :=  τ (S)↑X S ⊆ X (Filter of) Neighborhoods of S in (X, τ)
𝒩τ(x)  := 𝒩(x)  :=  τ (x)↑X x ∈ X (Filter of) Neighborhoods of x in (X, τ)
𝒩τ : X → Filter(X)    is the map    x ↦ 𝒩τ(x) Map of neighborhood filters (induced by τ) from X

Finer, coarser, subordinate

The following definition allows for the filter equivalent of "subsequences."[7]

  1. 𝒜 ≤ ℬ or ℬ ⊢ 𝒜, stated as ℬ is finer than 𝒜, 𝒜 is coarser than ℬ, or ℬ is subordinate to 𝒜:[8] If every A ∈ 𝒜 contains some B ∈ ℬ. That is, if for every A ∈ 𝒜, there is some B ∈ ℬ such that B ⊆ A.
      If 𝒜 ⊆ ℬ then 𝒜 ≤ ℬ. Thus ℬ ≤ ℘(X) is always true, for any ℬ ⊆ ℘(X).
    • 𝒜 ≤ ℬ if and only if 𝒜 ⊆ ℬ↑X.
    • If ℬ is upward closed (i.e. isotone), then 𝒜 ≤ ℬ if and only if 𝒜 ⊆ ℬ.[2]

The relation ≤ is not antisymmetric; that is, 𝒜 ≤ ℬ and ℬ ≤ 𝒜 does not necessarily imply ℬ = 𝒜; not even if both 𝒜 and ℬ are filter bases.[8] However, ≤ is transitive and reflexive so we may define an equivalence relation.

  1. 𝒜 is equivalent to ℬ: If 𝒜 ≤ ℬ and ℬ ≤ 𝒜.
    • Two families of sets 𝒜 and ℬ are equivalent if and only if their upward closures are equal.[2]
  2. 𝒜 ◅ ℬ or ℬ ▻ 𝒜 stated as 𝒜 is a refinement of ℬ, 𝒜 refines ℬ: If every A ∈ 𝒜 is contained in some B ∈ ℬ. That is, if for every A ∈ 𝒜, there is some B ∈ ℬ such that A ⊆ B.
    • 𝒜 ◅ ℬ if and only if (X ∖ 𝒜) ≤ (X ∖ ℬ).[2]
  3. 𝒜 # ℬ, stated as 𝒜 and ℬ mesh:[3] if A ∩ B ≠ ∅ for all A ∈ 𝒜 and B ∈ ℬ.
    • If S ⊆ X then we write S # ℬ to mean { S } # ℬ.
  4. 𝒜 and ℬ are dissociated:[3] if 𝒜 and ℬ do not mesh.

Filters and filter bases

We now define properties that a collection ℬ ⊆ ℘(X) may have.

Definitions: We say that ℬ is/has the:
  1. Proper: ∅ ∉ ℬ.
  2. Degenerate: ∅ ∈ ℬ.
  3. Closed under (finite) intersections: If B, C ∈ ℬ then B ∩ C ∈ ℬ.
  4. Directed by (superset/reverse) inclusion:[5] If B, C ∈ ℬ then there exists some A ∈ ℬ such that A ⊆ B ∩ C.
    • Equivalently, ℬ is a directed set when we define B ≤ A if and only if A ⊆ B for all subsets A and B of X.
  5. Finite intersection property and is centralized:[9] The intersection of any finite collection of sets in ℬ is not empty. That is, if B1, ..., Bn ∈ ℬ then ∅ ≠ B1 ∩ ⋅⋅⋅ ∩ Bn.
  6. Upward closed/Isotone[2]: If ℬ = ℬ↑X, or equivalently, if whenever B ∈ ℬ and C is a set such that B ⊆ C ⊆ X then C ∈ ℬ.
    • ℬ↑X is the a unique smallest isotone collection of subsets of X, in which case we may say that ℬ↑X is generated by ℬ.
  7. Ultra: For any S ⊆ X there exists some B ∈ ℬ such that S ⊆ B or S ⊆ X ∖ B.

We now define special categories of collections ℬ ⊆ ℘(X).

Definitions: We say that ℬ is/is a(n):
  1. Dual ideal:[10] If ℬ ≠ ∅ is closed under (finite) intersections and upward closed.
  2. Filter:[3] if ℬ ≠ ∅[10] is proper, closed under (finite) intersections, and upward closed. Equivalently, a filter is a proper dual ideal.
    • The intersection of any non-empty set 𝔽 of filters on X is a filter on X, called the infimum or greatest lower bound of 𝔽. In contrast, the least upper bound of a family of filters may fail to be a filter.
  3. Filter base[3]/Prefilter: if ℬ ≠ ∅ is proper and directed by superset inclusion.
    • If ℬ is a filter base then the upward closure ℬ↑X is the unique filter containing ℬ called the filter generated by ℬ. We say that a filter ℱ is generated by a filter base ℬ if ℱ = ℬ↑X, in which case we say that ℬ is a filter base for ℱ.
  4. Filter subbase:[3] if ℬ ≠ ∅ has the finite intersection property (which implies that ℬ is proper).
    • The collection of all finite intersections of subsets of a filter subbase ℬ is a filter base called the filter base generated by ℬ[3] and the filter generated by this filter base is called the filter generated by the filter subbase ℬ.
  5. Ultrafilter: if ℬ is ultra and a filter.
    • Every filter is equal to the intersection of all ultrafilters containing it.
  6. Free:[1] ker ℬ = ∅.
    • Every non-principal ultrafilter is free.
  7. Principal:[1] ker ℬ ∈ ℬ.
    • Every filter on a finite set is principal as is every cofinite filter on an infinite set.[1]
  8. Discrete/Principal at x ∈ X:[11] { x } = ker ℬ ∈ ℬ.
  9. Indiscrete:[11] ℬ = { X }
  10. Additive:[12] For every B ∈ ℬ, there exists some U ∈ ℬ such that U + U ⊆ B (this assumes that X is a group).
    • If ℬ is a filter then this happens if and only if ℬ ⊆ ℬ + ℬ.[12]
  11. Maximal:[3][13] For any filter base 𝒟 on X, if ℬ ≤ 𝒟 then 𝒟 ≤ ℬ. That is, if ℬ has no properly subordinate filter base.
    • A filter base ℬ on X is maximal if and only if for any S ⊆ X, either there exists some B ∈ ℬ such that B ⊆ S or else there exists some B ∈ ℬ such that X ∖ B ⊆ S.[3]
    • It may be shown using Zorn's lemma that given any filter base, there is a maximal filter base subordinate to it.[3]

Properties
Images and preimages of filter bases
  • If ℬ is a filter base on X then f (ℬ) is a filter base on Y, f -1 (f (ℬ)) is a filter base on X, and moreover, f -1 (f (ℬ)) ≤ ℬ.[4]
  • If ℬ is a filter on X then f (ℬ) is a filter base on Y, f (ℬ)↑Y := { S ⊆ Y : f -1 (S) ∈ ℬ }, and f (ℬ) is a filter on Y if and only if f is surjective.[14]
  • If ℬ is an ultrafilter on X and f is surjective then f (ℬ) is an ultrafilter on Y.[15]
  • If f is a bijection then ℬ is a filter base (resp. filter, ultrafilter) on X if and only if the same is true of f (ℬ) on Y.[15]
  • If 𝒞 is a filter base on Y then f -1 (𝒞) is a filter base on X if and only if ∅ ∉ f -1 (𝒞), in which case 𝒞 ≤ f (f -1 (𝒞)).[4]
  • If 𝒞 is a filter on Y then f -1 (𝒞) is a filter base on X but it may fail to be a filter on X even if f is surjective.[15]


Examples of filter bases and filters
  • If ℬ is a filter base then the trace of ℬ on S is a filter base if B ∩ S ≠ ∅ for all B ∈ ℬ.
  • If 𝒜 ≤ ℬ and ℬ is a filter base then 𝒜 and ℬ mesh[10] (see footnote for proof).[16]

Filter bases on topological spaces
Definition:[3] Say that a point x ∈ X is a cluster point of a collection of subsets ℬ of X if B ∩ N ≠ ∅ for every B ∈ ℬ and every neighborhood N of x in X. We denote the set of all cluster points of ℬ by cl ℬ.

Note that the set of all cluster points of a filter base cl ℬ in a topological space X is a closed subset of X and that moreover, cl ℬ = ∩ { cl B : B ∈ ℬ}.[3]

Note that if 𝒜 and ℬ are filters on X then 𝒜 < ℬ if and only if ℬ ⊆ 𝒜.

Definition:[3] Say that a collection of subsets ℬ of X converges to a point x ∈ X, that x is a limit of ℬ in X, and write ℬ → x in X if 𝒩(x) < ℬ. That is, if every neighborhood N of x contains some element of ℬ as a subset. We denote the set of all limit points of ℬ by lim ℬ.

Note that if x is a limit of a filter base ℬ in a topological space X, then x is a cluster point of ℬ.[3] If X is Hausdorff then a filter base on X has at most one limit point.[3]

Relation to topology

The relation ≤ is of fundamental importance to applying filters to topology. We may use the ≤ relation to define the analogue of "subsequence" for filter bases[7] and also to define convergence for filter bases. We will use these definitions to characterize in terms of filters and filter bases concepts like continuity and limits of functions.

  • The notion of "ℬ is subordinate to 𝒜" (written ℬ ⊢ 𝒜) is for filters and filter bases what "xn• = (xni)∞
    i=1     is a subsequence of x• = (xi)∞
    i=1    " is for sequences (and nets).[7]
    • Indeed, if we let 𝒳 := { x≥ i : i ∈ ℕ  } denotes the set of tails of x• and 𝒮 denotes the set of tails of the subsequence xn•, then 𝒮 ⊢ 𝒳 (i.e. 𝒳 ≤ 𝒮) is true but 𝒳 ⊢ 𝒮 is in general false.
  • If x• = (xi)i ∈ I is a net in a topological space X, 𝒳 := { x≥ i : i ∈ I  } is the set of its tails, and 𝒩(x) is the neighborhood filter at a point x ∈ X, then x• → x in X if and only if 𝒩(x) ≤ 𝒳.

Topological vector spaces definitions

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 addition. For this reason, we give definitions for an arbitrary commutative topological group with identity (X, +).

Canonical uniformity and Cauchy nets and filters

Let X be an additive commutative topological group with identity element 0.

Definition:[17] If S is a subset of an additive group G and N is a set containing 0, then we say that S is N-small or small of order N if S - S ⊆ N.
Definition:[18] If X is an additive commutative topological group with a neighborhood basis 𝒩 of the identity element 0, then the canonical uniformity on X is the uniform structure that has as a base of vicinities of the diagonal Δ in X × X all sets of the form Δ(N) := { (x, y) ∈ X × X : x - y ∈ U} as N ranges over 𝒩.

Having define a uniform structure on commutative topological groups, the notions of Cauchy nets, Cauchy filters, sequential completeness, and other notions are now defined via their usual definitions for uniform structures. However, for clarity, we review the relevant definitions again.

Cauchy nets
Definition:[17] A net x• = (xi)i ∈ I in X is called a Cauchy net if for every neighborhood N of 0 in X, there exists some i0 ∈ I such that xi - xj ∈ N for all i, j ≥ i0 where i, j ∈ I. A Cauchy sequence is a Cauchy net that is a sequence.

Cauchy filter bases
Definition:[17] A filter base ℬ on an additive topological group X called a Cauchy filter base if for every neighborhood N of 0 in X, there exists some B ∈ ℬ such that B - B ⊆ N.

The canonical uniformity is independent of the neighborhood basis 𝒩 that is chosen.

Complete topological group
Definition: A subset S of a topological group X is called complete if it satisfies any of the following equivalent conditions:
  1. S is a complete subset under the uniformity induced on S by the canonical uniformity;
  2. every Cauchy net in X that is contained in S converges to a point of S;
  3. every Cauchy filter in X that is contained in S converges to a point of S.
Definition: A topological group X is called complete if X is complete as a subset of itself.

Examples and sufficient conditions

Topologizing the set of filter bases and Top(X)

Starting with nothing more than a set X, one may topologize the set of ℙ := Prefilters(X) of all filter bases on X with the Stone topology. We first define and describe the basic properties of this topology and then show how one may use it to easily topologize the set of all topologies on X; something is not easily done with nets in X.

To reduce confusion we will adhere to the following notational conventions:

  • Lower case letters for elements x ∈ X.
  • Upper case letters for subsets S ⊆ X.
  • Upper case calligraphy letters for subsets 𝒜 ⊆ ℘(X).
  • Upper case double-struck letters for subsets ℙ ⊆ ℘(℘(X)).

Observe that if R ⊆ S ⊆ X then { 𝒜 ∈ ℘(℘(X)) : R ∈ 𝒜↑X } ⊆ { 𝒜 ∈ ℘(℘(X)) : S ∈ 𝒜↑X }. For every S ⊆ X, let

𝕆(S) := { 𝒜 ∈ ℙ : S ∈ 𝒜↑X }

where note that 𝕆(X) = ℙ and 𝕆(∅) = ∅. One may show that for all R, S ⊆ X the following holds:

𝕆(R ∩ S) = 𝕆(R) ∩ 𝕆(S) ⊆ 𝕆(R) ∪ 𝕆(S) ⊆ 𝕆(R ∪ S)

where in particular, the equality 𝕆(R ∩ S) = 𝕆(R) ∩ 𝕆(S) shows that the collection of sets { 𝕆(S) : S ⊆ X } form a basis for a topology on ℙ, which we will henceforth assume ℙ carries. We will assume that any subset of ℙ carries the subspace topology.

Recall that every τ ∈ Top(X) induces a canonical map 𝒩τ : X → Filter(X) defined by x ↦ 𝒩τ(x). Clearly, 𝒩τ : X → Filter(X) is injective if and only if τ is T0 (i.e. a Kolmogorov space). Let 𝒩• : Top(X) → Func(X;  ℙ) denote the map τ ↦ 𝒩τ. Since 𝒩• : Top(X) → Func(X;  ℙ) is clearly injective, to define a topology on Top(X) it suffices to define a topology on the range Im 𝒩• := { 𝒩τ : τ ∈ Top(X) }. So endow Func(X;  ℙ) with the topology of pointwise convergence (no topology on X is needed to do this) and endow Im 𝒩• with the subspace topology. We've thus topologized Top(X).

We now describe some additional properties of the Stone topology. For any 𝕊 ⊆ ℙ and 𝒜 ∈ ℙ,

  • 𝒜 belongs to the closure of 𝕊 in ℙ if and only if 𝒜 ⊆ ∪𝒮 ∈ 𝕊 𝒮↑X.
  • 𝕊 is a neighborhood of 𝒜 in ℙ if and only if there exists some A ∈ 𝒜 such that 𝕆(A) = { 𝒫 ∈ ℙ : A ∈ 𝒫↑X } ⊆ 𝕊 (i.e. for all 𝒫 ∈ ℙ, if A ∈ 𝒫↑X then 𝒫 ∈ 𝕊).

For every τ ∈ Top(X), the map 𝒩τ : (X, τ) → Im 𝒩τ is continuous, closed, and open (where Im 𝒩τ has the subspace topology inherited from ℙ). In addition, if 𝔉 : X → Filter(X) is a map such that x ∈ ker 𝔉(x) = ∩F ∈ 𝔉(x) F for every x ∈ X, then for every x ∈ X and every F ∈ 𝔉(x), 𝔉(F) is a neighborhood of 𝔉(x) in Im 𝔉 (where Im 𝔉 has the subspace topology inherited from ℙ).

See also

References
  1. Dolecki 2016, pp. 33-35.
  2. Dolecki 2016, pp. 27-29.
  3. Narici 2011, pp. 2-7.
  4. Dugundji 1966, pp. 215-221.
  5. Wilansky 2013, p. 5.
  6. Dolecki 2016, p. 10.
  7. Dugundji 1966, p. 212.
  8. Narici 2011, pp. 3-4.
  9. Arkhangelʹskiĭ 1984, pp. 7-8.
  10. Dugundji 1966, pp. 211-213.
  11. Wilansky 2013, pp. 44-46.
  12. Wilansky 2013, pp. 40-46.
  13. Dugundji 1966, pp. 218-220.
  14. Dolecki 2016, pp. 37-39.
  15. Arkhangelʹskiĭ 1984, pp. 20-22.
  16. Suppose A ∈ 𝒜 and B ∈ ℬ were such that A ∩ B = ∅. Since 𝒜 ≤ ℬ there exists some C ∈ ℬ such that C ⊆ A so that C ∩ B ⊆ A ∩ B = ∅, contradicting the fact that ℬ is a filter base. ∎
  17. Narici 2011, p. 48.
  18. Edwards 1995, p. 61.
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