In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Pseudometrics and metrics

A pseudometric on a set is a map satisfying the following properties:

  1. ;
  2. Symmetry: ;
  3. Subadditivity:

A pseudometric is called a metric if it satisfies:

  1. Identity of indiscernibles: for all if then

Ultrapseudometric

A pseudometric on is called a ultrapseudometric or a strong pseudometric if it satisfies:

  1. Strong/Ultrametric triangle inequality:

Pseudometric space

A pseudometric space is a pair consisting of a set and a pseudometric on such that 's topology is identical to the topology on induced by We call a pseudometric space a metric space (resp. ultrapseudometric space) when is a metric (resp. ultrapseudometric).

Topology induced by a pseudometric

If is a pseudometric on a set then collection of open balls:

as ranges over and ranges over the positive real numbers, forms a basis for a topology on that is called the -topology or the pseudometric topology on induced by

Convention: If is a pseudometric space and is treated as a topological space, then unless indicated otherwise, it should be assumed that is endowed with the topology induced by

Pseudometrizable space

A topological space is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) on such that is equal to the topology induced by [1]

Pseudometrics and values on topological groups

An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.

A topology on a real or complex vector space is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes into a topological vector space).

Every topological vector space (TVS) is an additive commutative topological group but not all group topologies on are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space may fail to make scalar multiplication continuous. For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

Translation invariant pseudometrics

If is an additive group then we say that a pseudometric on is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

  1. Translation invariance: ;

Value/G-seminorm

If is a topological group the a value or G-seminorm on (the G stands for Group) is a real-valued map with the following properties:[2]

  1. Non-negative:
  2. Subadditive: ;
  3. Symmetric:

where we call a G-seminorm a G-norm if it satisfies the additional condition:

  1. Total/Positive definite: If then

Properties of values

If is a value on a vector space then:

Equivalence on topological groups

Theorem[2] — Suppose that is an additive commutative group. If is a translation invariant pseudometric on then the map is a value on called the value associated with , and moreover, generates a group topology on (i.e. the -topology on makes into a topological group). Conversely, if is a value on then the map is a translation-invariant pseudometric on and the value associated with is just

Pseudometrizable topological groups

Theorem[2] — If is an additive commutative topological group then the following are equivalent:

  1. is induced by a pseudometric; (i.e. is pseudometrizable);
  2. is induced by a translation-invariant pseudometric;
  3. the identity element in has a countable neighborhood basis.

If is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric." A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable.

An invariant pseudometric that doesn't induce a vector topology

Let be a non-trivial (i.e. ) real or complex vector space and let be the translation-invariant trivial metric on defined by and such that The topology that induces on is the discrete topology, which makes into a commutative topological group under addition but does not form a vector topology on because is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on

This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.

Additive sequences

A collection of subsets of a vector space is called additive[5] if for every there exists some such that

Continuity of addition at 0 — If is a group (as all vector spaces are), is a topology on and is endowed with the product topology, then the addition map (i.e. the map ) is continuous at the origin of if and only if the set of neighborhoods of the origin in is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."[5]

All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.

Theorem — Let be a collection of subsets of a vector space such that and for all For all let

Define by if and otherwise let

Then is subadditive (meaning ) and on so in particular If all are symmetric sets then and if all are balanced then for all scalars such that and all If is a topological vector space and if all are neighborhoods of the origin then is continuous, where if in addition is Hausdorff and forms a basis of balanced neighborhoods of the origin in then is a metric defining the vector topology on

Proof

Assume that always denotes a finite sequence of non-negative integers and use the notation:

For any integers and

From this it follows that if consists of distinct positive integers then

It will now be shown by induction on that if consists of non-negative integers such that for some integer then This is clearly true for and so assume that which implies that all are positive. If all are distinct then this step is done, and otherwise pick distinct indices such that and construct from by replacing each with and deleting the element of (all other elements of are transferred to unchanged). Observe that and (because ) so by appealing to the inductive hypothesis we conclude that as desired.

It is clear that and that so to prove that is subadditive, it suffices to prove that when are such that which implies that This is an exercise. If all are symmetric then if and only if from which it follows that and If all are balanced then the inequality for all unit scalars such that is proved similarly. Because is a nonnegative subadditive function satisfying as described in the article on sublinear functionals, is uniformly continuous on if and only if is continuous at the origin. If all are neighborhoods of the origin then for any real pick an integer such that so that implies If the set of all form basis of balanced neighborhoods of the origin then it may be shown that for any there exists some such that implies

Paranorms

If is a vector space over the real or complex numbers then a paranorm on is a G-seminorm (defined above) on that satisfies any of the following additional conditions, each of which begins with "for all sequences in and all convergent sequences of scalars ":[6]

  1. Continuity of multiplication: if is a scalar and are such that and then
  2. Both of the conditions:
    • if and if is such that then ;
    • if then for every scalar
  3. Both of the conditions:
    • if and for some scalar then ;
    • if then
  4. Separate continuity:[7]
    • if for some scalar then for every ;
    • if is a scalar, and then .

A paranorm is called total if in addition it satisfies:

Properties of paranorms

If is a paranorm on a vector space then the map defined by is a translation-invariant pseudometric on that defines a vector topology on [8]

If is a paranorm on a vector space then:

Examples of paranorms

F-seminorms

If is a vector space over the real or complex numbers then an F-seminorm on (the stands for Fréchet) is a real-valued map with the following four properties: [11]

  1. Non-negative:
  2. Subadditive: for all
  3. Balanced: for all scalars satisfying
    • This condition guarantees that each set of the form or for some is a balanced set.
  4. For every as
    • The sequence can be replaced by any positive sequence converging to the zero.[12]

An F-seminorm is called an F-norm if in addition it satisfies:

  1. Total/Positive definite: implies

An F-seminorm is called monotone if it satisfies:

  1. Monotone: for all non-zero and all real and such that [12]

F-seminormed spaces

An F-seminormed space (resp. F-normed space)[12] is a pair consisting of a vector space and an F-seminorm (resp. F-norm) on

If and are F-seminormed spaces then a map is called an isometric embedding[12] if

Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.[12]

Examples of F-seminorms

Properties of F-seminorms

Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.[7] Every F-seminorm on a vector space is a value on In particular, and for all

Topology induced by a single F-seminorm

Theorem[11] — Let be an F-seminorm on a vector space Then the map defined by is a translation invariant pseudometric on that defines a vector topology on If is an F-norm then is a metric. When is endowed with this topology then is a continuous map on

The balanced sets as ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. Similarly, the balanced sets as ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets.

Topology induced by a family of F-seminorms

Suppose that is a non-empty collection of F-seminorms on a vector space and for any finite subset and any let

The set forms a filter base on that also forms a neighborhood basis at the origin for a vector topology on denoted by [12] Each is a balanced and absorbing subset of [12] These sets satisfy[12]

Fréchet combination

Suppose that is a family of non-negative subadditive functions on a vector space

The Fréchet combination[8] of is defined to be the real-valued map

As an F-seminorm

Assume that is an increasing sequence of seminorms on and let be the Fréchet combination of Then is an F-seminorm on that induces the same locally convex topology as the family of seminorms.[13]

Since is increasing, a basis of open neighborhoods of the origin consists of all sets of the form as ranges over all positive integers and ranges over all positive real numbers.

The translation invariant pseudometric on induced by this F-seminorm is

This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.[14]

As a paranorm

If each is a paranorm then so is and moreover, induces the same topology on as the family of paranorms.[8] This is also true of the following paranorms on :

Generalization

The Fréchet combination can be generalized by use of a bounded remetrization function.

A bounded remetrization function[15] is a continuous non-negative non-decreasing map that has a bounded range, is subadditive (meaning that for all ), and satisfies if and only if

Examples of bounded remetrization functions include and [15] If is a pseudometric (respectively, metric) on and is a bounded remetrization function then is a bounded pseudometric (respectively, bounded metric) on that is uniformly equivalent to [15]

Suppose that is a family of non-negative F-seminorm on a vector space is a bounded remetrization function, and is a sequence of positive real numbers whose sum is finite. Then

defines a bounded F-seminorm that is uniformly equivalent to the [16] It has the property that for any net in if and only if for all [16] is an F-norm if and only if the separate points on [16]

Characterizations

Of (pseudo)metrics induced by (semi)norms

A pseudometric (resp. metric) is induced by a seminorm (resp. norm) on a vector space if and only if is translation invariant and absolutely homogeneous, which means that for all scalars and all in which case the function defined by is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by is equal to

Of pseudometrizable TVS

If is a topological vector space (TVS) (where note in particular that is assumed to be a vector topology) then the following are equivalent:[11]

  1. is pseudometrizable (i.e. the vector topology is induced by a pseudometric on ).
  2. has a countable neighborhood base at the origin.
  3. The topology on is induced by a translation-invariant pseudometric on
  4. The topology on is induced by an F-seminorm.
  5. The topology on is induced by a paranorm.

Of metrizable TVS

If is a TVS then the following are equivalent:

  1. is metrizable.
  2. is Hausdorff and pseudometrizable.
  3. is Hausdorff and has a countable neighborhood base at the origin.[11][12]
  4. The topology on is induced by a translation-invariant metric on [11]
  5. The topology on is induced by an F-norm.[11][12]
  6. The topology on is induced by a monotone F-norm.[12]
  7. The topology on is induced by a total paranorm.

Birkhoff–Kakutani theorem — If is a topological vector space then the following three conditions are equivalent:[17][note 1]

  1. The origin is closed in and there is a countable basis of neighborhoods for in
  2. is metrizable (as a topological space).
  3. There is a translation-invariant metric on that induces on the topology which is the given topology on

By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.

Of locally convex pseudometrizable TVS

If is TVS then the following are equivalent:[13]

  1. is locally convex and pseudometrizable.
  2. has a countable neighborhood base at the origin consisting of convex sets.
  3. The topology of is induced by a countable family of (continuous) seminorms.
  4. The topology of is induced by a countable increasing sequence of (continuous) seminorms (increasing means that for all
  5. The topology of is induced by an F-seminorm of the form:
    where are (continuous) seminorms on [18]

Quotients

Let be a vector subspace of a topological vector space

Examples and sufficient conditions

If is Hausdorff locally convex TVS then with the strong topology, is metrizable if and only if there exists a countable set of bounded subsets of such that every bounded subset of is contained in some element of [22]

The strong dual space of a metrizable locally convex space (such as a Fréchet space[23]) is a DF-space.[24] The strong dual of a DF-space is a Fréchet space.[25] The strong dual of a reflexive Fréchet space is a bornological space.[24] The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.[26] If is a metrizable locally convex space then its strong dual has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.[26]

Normability

A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin. Moreover, a TVS is normable if and only if it is Hausdorff and seminormable.[14] Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is not normable must be infinite dimensional.

If is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then is normable.[27]

If is a Hausdorff locally convex space then the following are equivalent:

  1. is normable.
  2. has a (von Neumann) bounded neighborhood of the origin.
  3. the strong dual space of is normable.[28]

and if this locally convex space is also metrizable, then the following may be appended to this list:

  1. the strong dual space of is metrizable.[28]
  2. the strong dual space of is a Fréchet–Urysohn locally convex space.[23]

In particular, if a metrizable locally convex space (such as a Fréchet space) is not normable then its strong dual space is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space is also neither metrizable nor normable.

Another consequence of this is that if is a reflexive locally convex TVS whose strong dual is metrizable then is necessarily a reflexive Fréchet space, is a DF-space, both and are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover, is normable if and only if is normable if and only if is Fréchet–Urysohn if and only if is metrizable. In particular, such a space is either a Banach space or else it is not even a Fréchet–Urysohn space.

Metrically bounded sets and bounded sets

Suppose that is a pseudometric space and The set is metrically bounded or -bounded if there exists a real number such that for all ; the smallest such is then called the diameter or -diameter of [14] If is bounded in a pseudometrizable TVS then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.[14]

Properties of pseudometrizable TVS

Theorem[29] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.

Completeness

Main article: Complete topological vector space

Every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it. If is a metrizable TVS and is a metric that defines 's topology, then its possible that is complete as a TVS (i.e. relative to its uniformity) but the metric is not a complete metric (such metrics exist even for ). Thus, if is a TVS whose topology is induced by a pseudometric then the notion of completeness of (as a TVS) and the notion of completeness of the pseudometric space are not always equivalent. The next theorem gives a condition for when they are equivalent:

Theorem — If is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric then is a complete pseudometric on if and only if is complete as a TVS.[36]

Theorem[37][38] (Klee) — Let be any[note 2] 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.

Theorem — If is a TVS whose topology is induced by a paranorm then is complete if and only if for every sequence in if then converges in [39]

If is a closed vector subspace of a complete pseudometrizable TVS then the quotient space is complete.[40] If is a complete vector subspace of a metrizable TVS and if the quotient space is complete then so is [40] If is not complete then but not complete, vector subspace of

A Baire separable topological group is metrizable if and only if it is cosmic.[23]

Subsets and subsequences

Banach-Saks theorem[45] — If is a sequence in a locally convex metrizable TVS that converges weakly to some then there exists a sequence in such that in and each is a convex combination of finitely many

Mackey's countability condition[14] — Suppose that is a locally convex metrizable TVS and that is a countable sequence of bounded subsets of Then there exists a bounded subset of and a sequence of positive real numbers such that for all

Generalized series

As described in this article's section on generalized series, for any -indexed family family of vectors from a TVS it is possible to define their sum as the limit of the net of finite partial sums where the domain is directed by If and for instance, then the generalized series converges if and only if converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series converges in a metrizable TVS, then the set is necessarily countable (that is, either finite or countably infinite);[proof 1] in other words, all but at most countably many will be zero and so this generalized series is actually a sum of at most countably many non-zero terms.

Linear maps

If is a pseudometrizable TVS and maps bounded subsets of to bounded subsets of then is continuous.[14] Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.[46] Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[46]

If is a linear map between TVSs and is metrizable then the following are equivalent:

  1. is continuous;
  2. is a (locally) bounded map (that is, maps (von Neumann) bounded subsets of to bounded subsets of );[12]
  3. is sequentially continuous;[12]
  4. the image under of every null sequence in is a bounded set[12] where by definition, a null sequence is a sequence that converges to the origin.
  5. maps null sequences to null sequences;

Open and almost open maps

Theorem: If is a complete pseudometrizable TVS, is a Hausdorff TVS, and is a closed and almost open linear surjection, then is an open map.[47]
Theorem: If is a surjective linear operator from a locally convex space onto a barrelled space (e.g. every complete pseudometrizable space is barrelled) then is almost open.[47]
Theorem: If is a surjective linear operator from a TVS onto a Baire space then is almost open.[47]
Theorem: Suppose is a continuous linear operator from a complete pseudometrizable TVS into a Hausdorff TVS If the image of is non-meager in then is a surjective open map and is a complete metrizable space.[47]

Hahn-Banach extension property

Main article: Hahn-Banach theorem

A vector subspace of a TVS has the extension property if any continuous linear functional on can be extended to a continuous linear functional on [22] Say that a TVS has the Hahn-Banach extension property (HBEP) if every vector subspace of has the extension property.[22]

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

Theorem (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.[22]

If a vector space has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.[22]

See also

Notes

  1. ^ In fact, this is true for topological group, for the proof doesn't use the scalar multiplications.
  2. ^ Not assumed to be translation-invariant.

Proofs

  1. ^ Suppose the net converges to some point in a metrizable TVS where recall that this net's domain is the directed set Like every convergent net, this convergent net of partial sums is a Cauchy net, which for this particular net means (by definition) that for every neighborhood of the origin in there exists a finite subset of such that for all finite supersets this implies that for every (by taking and ). Since is metrizable, it has a countable neighborhood basis at the origin, whose intersection is necessarily (since is a Hausdorff TVS). For every positive integer pick a finite subset such that for every If belongs to then belongs to Thus for every index that does not belong to the countable set

References

  1. ^ Narici & Beckenstein 2011, pp. 1–18.
  2. ^ a b c Narici & Beckenstein 2011, pp. 37–40.
  3. ^ a b Swartz 1992, p. 15.
  4. ^ Wilansky 2013, p. 17.
  5. ^ a b Wilansky 2013, pp. 40–47.
  6. ^ Wilansky 2013, p. 15.
  7. ^ a b Schechter 1996, pp. 689–691.
  8. ^ a b c d e f g h i j k l m n o Wilansky 2013, pp. 15–18.
  9. ^ a b c d Schechter 1996, p. 692.
  10. ^ a b Schechter 1996, p. 691.
  11. ^ a b c d e f g h i j k l Narici & Beckenstein 2011, pp. 91–95.
  12. ^ a b c d e f g h i j k l m n o p q r s t Jarchow 1981, pp. 38–42.
  13. ^ a b Narici & Beckenstein 2011, p. 123.
  14. ^ a b c d e f g h Narici & Beckenstein 2011, pp. 156–175.
  15. ^ a b c Schechter 1996, p. 487.
  16. ^ a b c Schechter 1996, pp. 692–693.
  17. ^ Köthe 1983, section 15.11
  18. ^ Schechter 1996, p. 706.
  19. ^ Narici & Beckenstein 2011, pp. 115–154.
  20. ^ Wilansky 2013, pp. 15–16.
  21. ^ Schaefer & Wolff 1999, pp. 91–92.
  22. ^ a b c d e Narici & Beckenstein 2011, pp. 225–273.
  23. ^ a b c d Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
  24. ^ a b Schaefer & Wolff 1999, p. 154.
  25. ^ Schaefer & Wolff 1999, p. 196.
  26. ^ a b c Schaefer & Wolff 1999, p. 153.
  27. ^ Schaefer & Wolff 1999, pp. 68–72.
  28. ^ a b Trèves 2006, p. 201.
  29. ^ Wilansky 2013, p. 57.
  30. ^ Jarchow 1981, p. 222.
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  32. ^ Narici & Beckenstein 2011, pp. 459–483.
  33. ^ Köthe 1969, p. 168.
  34. ^ Wilansky 2013, p. 59.
  35. ^ a b Schaefer & Wolff 1999, pp. 12–35.
  36. ^ Narici & Beckenstein 2011, pp. 47–50.
  37. ^ Schaefer & Wolff 1999, p. 35.
  38. ^ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
  39. ^ Wilansky 2013, pp. 56–57.
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  41. ^ Schaefer & Wolff 1999, pp. 190–202.
  42. ^ Narici & Beckenstein 2011, pp. 172–173.
  43. ^ a b Rudin 1991, p. 22.
  44. ^ Narici & Beckenstein 2011, pp. 441–457.
  45. ^ Rudin 1991, p. 67.
  46. ^ a b Narici & Beckenstein 2011, p. 125.
  47. ^ a b c d Narici & Beckenstein 2011, pp. 466–468.

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