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: ; Subadditivity:
A pseudometric is called a metric if it satisfies: Identity of indiscernibles: for all if then
Ultrapseudometric
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: 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
: 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: 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: Non-negative: Subadditive: ;
Symmetric:
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: and for all and positive integers The set is an additive subgroup of
Equivalence on topological groups
Pseudometrizable topological groups
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 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 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 if for every there exists some such that
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.
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 ": 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 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 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 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 <li>
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: Non-negative: Subadditive: for all Balanced: for all scalars satisfying This condition guarantees that each set of the form or for some is a balanced set. For every as The sequence can be replaced by any positive sequence converging to the zero.
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
F-seminormed spaces
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
Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.
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
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 Each is a balanced and absorbing subset of These sets satisfy
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
Fréchet combination
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.
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.
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 :
Generalization