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:
- Symmetry: ;
A pseudometric is called a metric if it satisfies:
- Identity of indiscernibles: for all if then
A pseudometric on is called a ultrapseudometric or a strong pseudometric if it satisfies:
- 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:
- 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 
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:
- Subadditive: ;
where we call a G-seminorm a G-norm if it satisfies the additional condition:
- Total/Positive definite: If then
Properties of values
If is a value on a vector space then:
Equivalence on topological groups
Theorem — 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
- is induced by a pseudometric; (i.e. is pseudometrizable);
- is induced by a translation-invariant pseudometric;
- 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.
A collection of subsets of a vector space is called additive 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."
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
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
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 ":
- Continuity of multiplication: if is a scalar and are such that and then
- Both of the conditions:
- if and if is such that then ;
- if then for every scalar
- Both of the conditions:
- if and for some scalar then ;
- if then
- Separate continuity:
- if for some scalar then for every ;
- if is a scalar, and then .
A paranorm is called total if in addition it satisfies:
- Total/Positive definite: implies
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 
If is a paranorm on a vector space then:
- the set is a vector subspace of 
- with 
- If a paranorm satisfies and scalars then is absolutely homogeneity (i.e. equality holds) and thus is a seminorm.
Examples of paranorms
- If is a translation-invariant pseudometric on a vector space that induces a vector topology on (i.e. is a TVS) then the map defines a continuous paranorm on ; moreover, the topology that this paranorm defines in is 
- If is a paranorm on then so is the map 
- Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
- Every seminorm is a paranorm.
- The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).
- The sum of two paranorms is a paranorm.
- If and are paranorms on then so is Moreover, and This makes the set of paranorms on into a conditionally complete lattice.
- Each of the following real-valued maps are paranorms on :
- The real-valued maps and are not a paranorms on 
- If is a Hamel basis on a vector space then the real-valued map that sends (where all but finitely many of the scalars are 0) to is a paranorm on which satisfies for all and scalars 
- The function is a paranorm on that is not balanced but nevertheless equivalent to the usual norm on Note that the function is subadditive.
- Let be a complex vector space and let denote considered as a vector space over Any paranorm on is also a paranorm on 
- Subadditive: ;
- Balanced: for all and all scalars satisfying ;
- This condition guarantees that each set of the form or for some is balanced.
- for every as
- The sequence can be replaced by any positive sequence converging to 0.
An F-seminorm is called an F-norm if in addition it satisfies:
- Total/Positive definite: implies
An F-seminorm is called monotone if it satisfies:
- Monotone: for all non-zero and all real and such that 
An F-seminormed space (resp. F-normed space) 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 if
Examples of F-seminorms
- Every positive scalar multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm).
- The sum of finitely many F-seminorms (resp. F-norms) is an F-seminorm (resp. F-norm).
- If and are F-seminorms on then so is their pointwise supremum The same is true of the supremum of any non-empty finite family of F-seminorms on 
- The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).
- A non-negative real-valued function on is a seminorm if and only if it is a convex F-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm. In particular, every seminorm is an F-seminorm.
- For any the map on defined by
is an F-norm that is not a norm.
- If is a linear map and if is an F-seminorm on then is an F-seminorm on 
- Let be a complex vector space and let denote considered as a vector space over Any F-seminorm on is also an F-seminorm on 
Properties of F-seminorms
Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm. Every F-seminorm on a vector space is a value on In particular, and for all
Topology induced by a single F-seminorm
Theorem — 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
- is the coarsest vector topology on making each continuous.
- is Hausdorff if and only if for every non-zero there exists some such that 
- If is the set of all continuous F-seminorms on then 
- If is the set of all pointwise suprema of non-empty finite subsets of of then is a directed family of F-seminorms and 
Suppose that is a family of non-negative subadditive functions on a vector space
The Fréchet combination 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.
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.
As a paranorm
If each is a paranorm then so is and moreover, induces the same topology on as the family of paranorms. This is also true of the following paranorms on :
The Fréchet combination can be generalized by use of a bounded remetrization function.
Examples of bounded remetrization functions include and  If is a pseudometric (resp. metric) on and is a bounded remetrization function then is a bounded pseudometric (resp. bounded metric) on that is uniformly equivalent to 
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
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
- is pseudometrizable (i.e. the vector topology is induced by a pseudometric on ).
- has a countable neighborhood base at the origin.
- The topology on is induced by a translation-invariant pseudometric on
- The topology on is induced by an F-seminorm.
- The topology on is induced by a paranorm.
Of metrizable TVS
If is a TVS then the following are equivalent:
- is metrizable.
- is Hausdorff and pseudometrizable.
- is Hausdorff and has a countable neighborhood base at the origin.
- The topology on is induced by a translation-invariant metric on 
- The topology on is induced by an F-norm.
- The topology on is induced by a monotone F-norm.
- The topology on is induced by a total paranorm.
- The origin is closed in and there is a countable basis of neighborhoods for in
- is metrizable (as a topological space).
- 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:
- is locally convex and pseudometrizable.
- has a countable neighborhood base at the origin consisting of convex sets.
- The topology of is induced by a countable family of (continuous) seminorms.
- The topology of is induced by a countable increasing sequence of (continuous) seminorms (increasing means that for all
- The topology of is induced by an F-seminorm of the form:
where are (continuous) seminorms on 
Let be a vector subspace of a topological vector space
- If is a pseudometrizable TVS then so is 
- If is a complete pseudometrizable TVS and is a closed vector subspace of then is complete.
- If is metrizable TVS and is a closed vector subspace of then is metrizable.
- If is an F-seminorm on then the map defined by
is an F-seminorm on that induces the usual quotient topology on  If in addition is an F-norm on and if is a closed vector subspace of then is an F-norm on 
Examples and sufficient conditions
- Every seminormed space is pseudometrizable with a canonical pseudometric given by for all .
- If is pseudometric TVS with a translation invariant pseudometric then defines a paranorm. However, if is a translation invariant pseudometric on the vector space (without the addition condition that is pseudometric TVS), then need not be either an F-seminorm nor a paranorm.
- If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.
- If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.
- Suppose is either a DF-space or an LM-space. If is a sequential space then it is either metrizable or else a Montel DF-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 
The strong dual space of a metrizable locally convex space (such as a Fréchet space) is a DF-space. The strong dual of a DF-space is a Fréchet space. The strong dual of a reflexive Fréchet space is a bornological space. The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space. 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.
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. 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 Hausdorff locally convex space then the following are equivalent:
- is normable.
- has a (von Neumann) bounded neighborhood of the origin.
- the strong dual space of is normable.
and if this locally convex space is also metrizable, then the following may be appended to this list:
- the strong dual space of is metrizable.
- the strong dual space of is a Fréchet–Urysohn locally convex space.
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  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.
Properties of pseudometrizable TVS
- Every metrizable locally convex TVS is a quasibarrelled space, bornological space, and a Mackey space.
- Every complete pseudometrizable TVS is a barrelled space and a Baire space (and hence non-meager). However, there exist metrizable Baire spaces that are not complete.
- If is a metrizable locally convex space, then the strong dual of is bornological if and only if it is barreled, if and only if it is infrabarreled.
- If is a complete pseudometrizable TVS and is a closed vector subspace of then is complete.
- The strong dual of a locally convex metrizable TVS is a webbed space.
- If and are complete metrizable TVSs (i.e. F-spaces) and if is coarser than then ; this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete. Said differently, if and are both F-spaces but with different topologies, then neither one of and contains the other as a subset. One particular consequence of this is, for example, that if is a Banach space and is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of (i.e. if or if for some constant ), then the only way that can be a Banach space (i.e. also be complete) is if these two norms and are equivalent; if they are not equivalent, then can not be a Banach space. As another consequence, if is a Banach space and is a Fréchet space, then the map is continuous if and only if the Fréchet space is the TVS (here, the Banach space is being considered as a TVS, which means that its norm is "forgetten" but its topology is remembered).
- A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.
- Any product of complete metrizable TVSs is a Baire space.
- A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension 
- A product of pseudometrizable TVSs is pseudometrizable if and only if it all but at most countably many of these TVSs have the trivial topology.
- Every complete pseudometrizable TVS is a barrelled space and a Baire space (and thus non-meager).
- The dimension of a complete metrizable TVS is either finite or uncountable.
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.
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 
If is a closed vector subspace of a complete pseudometrizable TVS then the quotient space is complete. If is a complete vector subspace of a metrizable TVS and if the quotient space is complete then so is  If is not complete then but not complete, vector subspace of
Subsets and subsequences
- Let be a separable locally convex metrizable topological vector space and let be its completion. If is a bounded subset of then there exists a bounded subset of such that 
- Every totally bounded subset of a locally convex metrizable TVS is contained in the closed convex balanced hull of some sequence in that converges to
- In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.
- If is a translation invariant metric on a vector space then for all and every positive integer 
- If is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence of positive real numbers diverging to such that 
- A subset of a complete metric space is closed if and only if it is complete. If a space is not complete, then is a closed subset of that is not complete.
- If is a metrizable locally convex TVS then for every bounded subset of there exists a bounded disk in such that and both and the auxiliary normed space induce the same subspace topology on 
Banach-Saks theorem — 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 — 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
If is a pseudometrizable TVS and maps bounded subsets of to bounded subsets of then is continuous. Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS. Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.
If is a linear map between TVSs and is metrizable then the following are equivalent:
- is continuous;
- is a (locally) bounded map (that is, maps (von Neumann) bounded subsets of to bounded subsets of );
- is sequentially continuous;
- the image under of every null sequence in is a bounded set where by definition, a null sequence is a sequence that converges to the origin.
- 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.
- 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.
- 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.
Hahn-Banach extension property
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  Say that a TVS has the Hahn-Banach extension property (HBEP) if every vector subspace of has the extension property.
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.
- Asymmetric norm – Generalization of the concept of a norm
- Complete metric space – Metric geometry
- Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
- Closed graph theorem (functional analysis) – Theorems for deducing continuity
- Equivalence of metrics
- F-space – Topological vector space with a complete translation-invariant metric
- Fréchet space – A locally convex topological vector space that is also a complete metric space
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Metric space – Mathematical set defining distance
- Open mapping theorem (functional analysis) – Condition for a linear operator to be open
- Pseudometric space – Generalization of metric spaces in mathematics
- Relation of norms and metrics
- Sublinear function
- Topological vector space – Vector space with a notion of nearness
- Uniform space – Topological space with a notion of uniform properties
- Ursescu theorem��– Generalization of closed graph, open mapping, and uniform boundedness theorem
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