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).
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[edit]
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.
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:
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
Theorem[2] — If is an additive commutative topological group then the following are equivalent:
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[edit]
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[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
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]
Continuity of multiplication: if is a scalar and are such that and then
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 [8]
Each of the following real-valued maps are paranorms on :
The real-valued maps and are not a paranorms on [8]
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 [8]
The function is a paranorm on that is not balanced but nevertheless equivalent to the usual norm on Note that the function is subadditive.[10]
Let be a complex vector space and let denote considered as a vector space over Any paranorm on is also a paranorm on [9]
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]
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 [12]
The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).[9]
A non-negative real-valued function on is a seminorm if and only if it is a convexF-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm.[10] 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 [12]
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 [9]
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
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.
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]
is the coarsest vector topology on making each continuous.[12]
is Hausdorff if and only if for every non-zero there exists some such that [12]
If is the set of all continuous F-seminorms on then [12]
If is the set of all pointwise suprema of non-empty finite subsets of of then is a directed family of F-seminorms and [12]
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.
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 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]
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