Metrizable topological vector space

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 formula_1 satisfying the following properties:
A pseudometric is called a metric if it satisfies: 
A pseudometric on is called a ultrapseudometric or a strong pseudometric if it satisfies: 
A pseudometric space is a pair formula_2 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 formula_2 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:
forms a basis for a topology on that is called the -topology or the pseudometric topology on induced by . 
A topological space formula_6 is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) on such that formula_7 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 (i.e. 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:
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 formula_8 with the following properties:
where we call a G-seminorm a G-norm if it satisfies the additional condition:
Properties of values.
If is a value on a vector space then:
An invariant pseudometric that doesn't induce a vector topology.
Let be a non-trivial (i.e. formula_9) real or complex vector space and let be the translation-invariant trivial metric on defined by formula_10 and formula_11 for all formula_12 such that formula_13 
The topology formula_7 that induces on is the discrete topology, which makes formula_6 into a commutative topological group under addition but does form a vector topology on because formula_6 is disconnected but every vector topology is connected. 
What fails is that scalar multiplication isn't continuous on formula_17 
This example shows that a translation-invariant (pseudo)metric is enough to guarantee a vector topology, which leads us to define paranorms and -seminorms.
Additive sequences.
A collection formula_18 of subsets of a vector space is called additive if for every formula_19 there exists some formula_20 such that formula_21
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. 
Assume that formula_22 always denotes a finite sequence of non-negative integers and use the notation: 
For any integers formula_25 and formula_26 
From this it follows that if formula_22 consists of distinct positive integers then formula_29 
It will now be shown by induction on formula_30 that if formula_22 consists of non-negative integers such that formula_32 for some integer formula_33 then formula_34 
This is clearly true for formula_35 and formula_36 so assume that formula_37 which implies that all formula_38 are positive. 
If all formula_38 are distinct then this step is done, and otherwise pick distinct indices formula_40 such that formula_41 and construct formula_42 from formula_43 by replacing each formula_38 with formula_45 and deleting the formula_46 element of formula_43 (all other elements of formula_43 are transferred to formula_49 unchanged). 
Observe that formula_50 and formula_51 (because formula_52) so by appealing to the inductive hypothesis we conclude that formula_53 as desired. 
It is clear that formula_54 and that formula_55 so to prove that formula_56 is subadditive, it suffices to prove that formula_57 when formula_12 are such that formula_59 which implies that formula_60 
This is an exercise. 
If all formula_61 are symmetric then formula_62 if and only if formula_63 from which it follows that formula_64 and formula_65 
If all formula_61 are balanced then the inequality formula_67 for all unit scalars formula_68 such that formula_69 is proved similarly. 
Because formula_56 is a nonnegative subadditive function satisfying formula_71 as described in the article on sublinear functionals, formula_56 is uniformly continuous on formula_73 if and only if formula_56 is continuous at the origin. 
If all formula_61 are neighborhoods of the origin then for any real formula_76 pick an integer formula_77 such that formula_78 so that formula_79 implies formula_80 
If the set of all formula_61 form basis of balanced neighborhoods of the origin then it may be shown that for any formula_82 there exists some formula_83 such that formula_84 implies formula_85 formula_86
Paranorms.
If is a vector space over the real or complex numbers then a paranorm on is a G-seminorm (defined above) formula_8 on that satisfies any of the following additional conditions, each of which begins with "for all sequences formula_88 in and all convergent sequences of scalars formula_89":
A paranorm is called total if in addition it satisfies:
Properties of paranorms.
If is a paranorm on a vector space then the map formula_1 defined by formula_91 is a translation-invariant pseudometric on that defines a vector topology on .
If is a paranorm on a vector space then:
"F"-seminorms.
If is a vector space over the real or complex numbers then an -seminorm on (the stands for Fréchet) is a real-valued map formula_8 with the following properties: 
An -seminorm is called an -norm if in addition it satisfies:
An -seminorm is called monotone if it satisfies:
"F"-seminormed spaces.
An -seminormed space (resp. -normed space) is a pair formula_93 consisting of a vector space and an -seminorm (resp. -norm) on . 
If formula_93 and formula_95 are -seminormed spaces then a map formula_96 is called an isometric embedding if formula_97 for all formula_98 
Every isometric embedding of one -seminormed space into another is a topological embedding, but the converse is not true in general. 
Properties of "F"-seminorms.
Every -seminorm is a paranorm and every paranorm is equivalent to some -seminorm. 
Every -seminorm on a vector space is a value on . In particular, formula_99 and formula_100 for all formula_101 
Topology induced by a family of "F"-seminorms.
Suppose that formula_102 is a non-empty collection of -seminorms on a vector space and for any finite subset formula_103 and any formula_76 let 
The set formula_106 forms a filter base on that also forms a neighborhood basis at the origin for a vector topology on denoted by formula_107 Each formula_108 is a balanced and absorbing subset of . These sets satisfy
Fréchet combination.
Suppose that formula_110 is a family of non-negative subadditive functions on a vector space . 
The Fréchet combination of formula_111 is defined to be the real-valued map 
As an "F"-seminorm.
Assume that formula_110 is an increasing sequence of seminorms on and let be the Fréchet combination of formula_114 
Then is an -seminorm on that induces the same locally convex topology as the family formula_111 of seminorms. 
Since formula_110 is increasing, a basis of open neighborhoods of the origin consists of all sets of the form formula_117 as ranges over all positive integers and formula_118 ranges over all positive real numbers. 
The translation invariant pseudometric on induced by this -seminorm is 
This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations. 
As a paranorm.
If each formula_120 is a paranorm then so is and moreover, induces the same topology on as the family formula_111 of paranorms. 
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 is a continuous non-negative non-decreasing map formula_122 that is subadditive (i.e. formula_123 for all formula_124 has a bounded range, and satisfies formula_125 if and only if formula_126 
Examples of bounded remetrization functions include formula_127 formula_128 formula_129 and formula_130 
If is a pseudometric (resp. metric) on and formula_131 is a bounded remetrization function then formula_132 is a bounded pseudometric (resp. bounded metric) on that is uniformly equivalent to . 
Suppose that formula_110 is a family of non-negative -seminorm on a vector space , } is a bounded remetrization function, and formula_134 is a sequence of positive real numbers whose sum is finite. 
Then 
defines a bounded -seminorm that is uniformly equivalent to the formula_114 
It has the property that for any net formula_137 in , formula_138 if and only if formula_139 for all formula_140 
formula_141 is an -norm if and only if the formula_111 separate points on .
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 formula_143 in which case the function defined by formula_144 is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by is equal to .
Of pseudometrizable TVS.
If formula_6 is a topological vector space (TVS) (where note in particular that formula_7 is assumed to be a vector topology) then the following are equivalent: 
Of metrizable TVS.
If formula_6 is a TVS then the following are equivalent:
Of locally convex pseudometrizable TVS.
If formula_6 is TVS then the following are equivalent: 
Quotients.
Let be a vector subspace of a topological vector space formula_17 
Examples and sufficient conditions.
If formula_73 is Hausdorff locally convex TVS then formula_73 with the strong topology, formula_152 is metrizable if and only if there exists a countable set formula_153 of bounded subsets of formula_73 such that every bounded subset of is contained in some element of formula_155
The strong dual space formula_156 of a metrizable locally convex space (such as a Fréchet space) formula_73 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 formula_73 is a metrizable locally convex space then its strong dual formula_156 has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled. 
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. 
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 normable must be infinite dimensional. 
If formula_160 is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then formula_160 is normable. 
If formula_73 is a Hausdorff locally convex space then the following are equivalent:
and if this locally convex space formula_73 is also metrizable, then the following may be appended to this list:
In particular, if a metrizable locally convex space formula_73 (such as a Fréchet space) is normable then its strong dual space formula_165 is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space formula_165 is also neither metrizable nor normable.
Another consequence of this is that if formula_73 is a reflexive locally convex TVS whose strong dual formula_165 is metrizable then formula_165 is necessarily a reflexive Fréchet space, formula_73 is a DF-space, both formula_73 and formula_165 are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover, formula_165 is normable if and only if formula_73 is normable if and only if formula_73 is Fréchet–Urysohn if and only if formula_73 is metrizable. In particular, such a space formula_73 is either a Banach space or else it is not even a Fréchet–Urysohn space.
Metrically bounded sets and bounded sets.
Suppose that formula_2 is a pseudometric space and formula_179 
The set is metrically bounded or -bounded if there exists a real number formula_180 such that formula_181 for all formula_182; 
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.
Completeness.
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 a complete metric (such metrics exist even for formula_183). 
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 formula_2 are not always equivalent. 
The next theorem gives a condition for when they are equivalent: 
If formula_160 is a closed vector subspace of a complete pseudometrizable TVS , then the quotient space formula_186 is complete. 
If formula_160 is a complete vector subspace of a metrizable TVS and if the quotient space formula_186 is complete then so is . If is not complete then formula_189 but not complete, vector subspace of . 
A Baire separable topological group is metrizable if and only if it is cosmic.
Linear maps.
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 formula_190 is a linear map between TVSs and is metrizable then the following are equivalent:
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:
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.