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 LMspace is an inductive limit of a sequence of locally convex metrizable TVS.
A pseudometric on a set is a map satisfying the following properties:
A pseudometric is called a metric if it satisfies:
Ultrapseudometric
A pseudometric on is called a ultrapseudometric or a strong pseudometric if it satisfies:
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).
If is a pseudometric on a set then collection of open balls:
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]}
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 nontrivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.
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:
If is a topological group the a value or Gseminorm on (the G stands for Group) is a realvalued map with the following properties:^{[2]}
where we call a Gseminorm a Gnorm if it satisfies the additional condition:
If is a value on a vector space then:
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 translationinvariant pseudometric on and the value associated with is just
Theorem^{[2]} — If is an additive commutative topological group then the following are equivalent:
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.
Let be a nontrivial (i.e. ) real or complex vector space and let be the translationinvariant 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 translationinvariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and Fseminorms.
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 nonnegative continuous realvalued 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 nonnegative 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 nonnegative 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 
If is a vector space over the real or complex numbers then a paranorm on is a Gseminorm (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]}
A paranorm is called total if in addition it satisfies:
If is a paranorm on a vector space then the map defined by is a translationinvariant pseudometric on that defines a vector topology on ^{[8]}
If is a paranorm on a vector space then:
If is a vector space over the real or complex numbers then an Fseminorm on (the stands for Fréchet) is a realvalued map with the following four properties: ^{[11]}
An Fseminorm is called an Fnorm if in addition it satisfies:
An Fseminorm is called monotone if it satisfies:
An Fseminormed space (resp. Fnormed space)^{[12]} is a pair consisting of a vector space and an Fseminorm (resp. Fnorm) on
If and are Fseminormed spaces then a map is called an isometric embedding^{[12]} if
Every isometric embedding of one Fseminormed space into another is a topological embedding, but the converse is not true in general.^{[12]}
Every Fseminorm is a paranorm and every paranorm is equivalent to some Fseminorm.^{[7]} Every Fseminorm on a vector space is a value on In particular, and for all
Theorem^{[11]} — Let be an Fseminorm on a vector space Then the map defined by is a translation invariant pseudometric on that defines a vector topology on If is an Fnorm 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.
Suppose that is a nonempty collection of Fseminorms 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]}
Suppose that is a family of nonnegative subadditive functions on a vector space
The Fréchet combination^{[8]} of is defined to be the realvalued map
Assume that is an increasing sequence of seminorms on and let be the Fréchet combination of Then is an Fseminorm 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 Fseminorm is
This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.^{[14]}
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 :
The Fréchet combination can be generalized by use of a bounded remetrization function.
A bounded remetrization function^{[15]} is a continuous nonnegative nondecreasing 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 nonnegative Fseminorm on a vector space is a bounded remetrization function, and is a sequence of positive real numbers whose sum is finite. Then
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
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]}
If is a TVS then the following are equivalent:
Birkhoff–Kakutani theorem — If is a topological vector space then the following three conditions are equivalent:^{[17]}^{[note 1]}
By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translationinvariant.
If is TVS then the following are equivalent:^{[13]}
Let be a vector subspace of a topological vector space
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 DFspace.^{[24]} The strong dual of a DFspace 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]}
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 finitedimensional vector space is a normable locally convex complete TVS, being TVSisomorphic 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:
and if this locally convex space is also metrizable, then the following may be appended to this list:
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 DFspace, 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.
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]}
Theorem^{[29]} — All infinitedimensional separable complete metrizable TVS are homeomorphic.
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 completeTVS.
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]}
BanachSaks 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 nonzero terms.
If is a pseudometrizable TVS and maps bounded subsets of to bounded subsets of then is continuous.^{[14]} Discontinuous linear functionals exist on any infinitedimensional pseudometrizable TVS.^{[46]} Thus, a pseudometrizable TVS is finitedimensional 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:
Open and almost open maps