Metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector spaces (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric).
Pseudometrics and metrics
A pseudometric on a set X is a map d : X × X → ℝ satisfying the following properties:
- d(x, x) = 0 for all x ∈ X;
- Symmetry: d(x, y) = d(y, x) for all x, y ∈ X;
- Subadditivity: d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
A pseudometric is called a metric if it satisfies:
- Identity of indiscernibles: for all x, y ∈ X, if d(x, y) = 0 then x = y.
- Ultrapseudometric
A pseudometric d on X is called a ultrapseudometric or a strong pseudometric if it satisfies:
- Strong/Ultrametric triangle inequality: for all x, y ∈ X, d(x, y) ≤ max { d(x, z), d(y, z)}.
- Pseudometric space
- Definition: A pseudometric space is a pair (X, d) consisting of a set X and a pseudometric d on X such that X's topology is identical to the topology on X induced by d. We call a pseudometric space (X, d) a metric space (resp. ultrapseudometric space) when d is a metric (resp. ultrapseudometric).
Topology induced by a pseudometric
If d is a pseudometric on a set X then collection of open balls:
- Br(z) := { x ∈ X : d(x, z) < r }, as z ranges over X and r ranges over the positive real numbers,
forms a basis for a topology on X that is called the d-topology or the pseudometric topology on X induced by d.
- Convention: If (X, d) is a pseudometric space and X is treated as a topological space, then unless indicated otherwise, it should be assumed that X is endowed with the topology induced by d.
- Pseudometrizable space
- Definition: A topological space (X, τ) is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) d on X such that τ is equal to the topology induced by d.[1]
Pseudometrics and values on topological groups
- Definition: An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.
- Definition: A topology τ on a real or complex vector space X 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 X into a topological vector space).
Every topological vector space (TVS) X is an additive commutative topological group but not all group topologies on X are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space X 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 X is an additive group then we say that a pseudometric d on X is translation invariant or just invariant if it satisfies any of the following equivalent conditions:
- Translation invariance: d(x + z, y + z) = d(x, y) for all x, y, z ∈ X;
- d(x, y) = d(x - y, 0) for all x, y ∈ X.
Value/G-seminorm
If X is a topological group the a value or G-seminorm on X (the G stands for Group) is a real-valued map p : X → ℝ with the following properties:[2]
- Non-negative: p ≥ 0.
- Subadditive: p(x+y) ≤ p(x) + p(y) for all x, y ∈ X;
- p(0) = 0.
- Symmetric: p(-x) = p(x) for all x ∈ X.
where we call a G-seminorm a G-norm if it satisfies the additional condition:
- Total/Positive definite: If p(x) = 0 then x = 0.
Properties of values
If p is a value on a vector space X then:
Equivalence on topological groups
Theorem[2] — Suppose that X is an additive commutative group. If d is a translation invariant pseudometric on X then the map p(x) := d(x, 0) is a value on X called the value associated with d, and moreover, d generates a group topology on X (i.e. the d-topology on X makes X into a topological group). Convergesly, if p is a value on X then the map {{{1}}} is a translation-invariant pseudometric on X and the value associated with d is just p.
Pseudometrizable topological groups
Theorem[2] — If (X, τ) is an additive commutative topological group then the following are equivalent:
- τ is induced by a pseudometric; (i.e. (X, τ) is pseudometrizable);
- τ is induced by a translation-invariant pseudometric;
- the identity element in (X, τ) has a countable neighborhood basis.
If (X, τ) 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 X be a non-trivial (i.e. X ≠ { 0 }) real or complex vector space and let d be the translation-invariant trivial metric on X defined by d(x, x) = 0 and d(x, y) = 1 for all x, y ∈ X such that x ≠ y. The topology τ that d induces on X is the discrete topology, which makes (X, τ) into a commutative topological group under addition but does not form a vector topology on X because (X, τ) is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on (X, τ).
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.
Paranorms
If X is a vector space over the real or complex numbers then a paranorm on X is a G-seminorm (defined above) p : X → ℝ on X that satisfies any of the following additional conditions, each of which begins with "for all sequences x• = (xi)∞
i=1 in X and all convergent sequences of scalars s• = (si)∞
i=1":[5]
- Continuity of multiplication: if s is a scalar and x ∈ X are such that p(xi - x) → 0 and s• → s, then p(si xi - sx) → 0.
- Both of the conditions:
- if s• → 0 and if x ∈ X is such that p(xi - x) → 0 then p(si xi) → 0;
- if p(xi) → 0 then p(s xi) → 0 for every scalar s.
- Both of the conditions:
- if p(xi) → 0 and s• → s for some scalar s then p(si xi) → 0;
- if s• → 0 then p(si x) → 0 for all x ∈ X.
- Separate continuity:[6]
- if s• → s for some scalar s then p(si x - sx) → 0 for every x ∈ X;
- if s is a scalar, x ∈ X, and p(xi - x) → 0 then p(s xi - sx) → 0.
A paranorm is called an total if in addition it satisfies:
- Total/Positive definite: p(x) = 0 implies x = 0.
Properties of paranorms
- If p is a paranorm on a vector space X then the map d : X × X → ℝ defined by d(x, y) := p(x - y) is a translation-invariant pseudometric on X that defines a vector topology on X.[7]
If p is a paranorm on a vector space X then:
Examples of paranorms
- If d is a translation-invariant pseudometric on a vector space X that induces a vector topology τ on X (i.e. (X, τ) is a TVS) then the map p(x) := d(x - y, 0) defines a continuous paranorm on (X, τ); moreover, the topology that this paranorm p defines in X is τ.[7]
- If p is a paranorm on X then so is the map q(x) := p(x)/[1 + p(x)].[7]
- Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
- Every seminorm is a paranorm.[7]
- The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).[8]
- The sum of two paranorms is a paranorm.[7]
- If p and q are paranorms on X then so is (p∧q)(x) := inf { p(y) + q(z) : x = y + z with y, z ∈ X }. Moreover, p∧q ≤ p and p∧q ≤ q. This makes the set of paranorms on X into a conditionally complete lattice.[7]
- Each of the following real-valued maps are paranorms on X := ℝ2:
- (x, y) ↦ |x|
- (x, y) ↦ |x| + |y|
- The real-valued map (x, y) ↦ √x2 + y2 is not paranorms on X := ℝ2.[7]
- If x• = (xi)i ∈ I is a Hamel basis on a vector space X then the real-valued map that sends x = ∑i ∈ I sixi ∈ X (where all but finitely many of the scalars si are 0) to ∑i ∈ I √|si| is a paranorm on X, which satisfies p(sx) = √|s| p(x) for all x ∈ X and scalars s.[7]
- The function p(x) := |sin(πx)| + min {2, |x| } is a paranorm on ℝ that is not balanced but nevertheless equivalent to the usual norm on R. Note that the function x ↦ |sin(πx)| is subadditive.[9]
- Let Xℂ be a complex vector space and let Xℝ denote Xℂ considered as a vctor space over ℝ. Any paranorm on Xℂ is also a paranorm on Xℝ.[8]
F-seminorms
If X is a vector space over the real or complex numbers then an F-seminorm on X (the F stands for Fréchet) is a real-valued map p : X → ℝ with the following properties:[10]
- Non-negative: p ≥ 0.
- Subadditive: p(x+y) ≤ p(x) + p(y) for all x, y ∈ X;
- Balanced: p(ax) ≤ p(x) for all x ∈ X and all scalars a satisfying |a| ≤ 1 ;
- This condition guarantees that each set of the form { x ∈ X : p(x) ≤ r } or { x ∈ X : p(x) < r } for some r ≥ 0 is balanced.
- for every x ∈ X, p(1/n x) → 0 as n → ∞
- The sequence (1/n)∞
n=1 can be replaced by any positive sequence converging to 0.[11]
- The sequence (1/n)∞
An F-seminorm is called an F-norm if in addition it satisfies:
- Total/Positive definite: p(x) = 0 implies x = 0.
An F-seminorm is called monotone if it satisfies:
- Monotone: p(rx) < p(sx) for all non-zero x ∈ X and all real s and t such that s < t.[11]
F-seminormed spaces
- Definition:[11] An F-seminormed space (resp. F-normed space) is a pair (X, p) consisting of a vector space X and an F-seminorm (resp. F-norm) p on X.
- Definition:[11] If (X, p) and (Z, q) are F-seminormed spaces then a map f : X → Z is called an isometric embedding if q(f (x) - f (y)) = p(x - y) for all x, y ∈ X.
Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.[11]
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 p and q are F-seminorms on X then so is their pointwise supremum x ↦ sup { p(x), q(x) }. The same is true of the supremum of any non-empty finite family of F-seminorms on X.[11]
- The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).[8]
- A non-negative real-valued function on X 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.[9]
- In particular, every seminorm is an F-seminorm.
- For any 0 < p < 1, the map f on ℝn defined by [f(x1, ..., xn)]p := |x1|p + ⋅⋅⋅ + |xn|p is an F-norm that is not a norm.
- If L : X → Y is a linear map and if q is an F-seminorm on Y, then q ∘ L is an F-seminorm on X.[11]
- Let Xℂ be a complex vector space and let Xℝ denote Xℂ considered as a vctor space over ℝ. Any F-seminorm on Xℂ is also an F-seminorm on Xℝ.[8]
Properties of F-seminorms
- Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.[12]
- Every F-seminorm on a vector space X is a value on X. In particular,
- p(0) = 0;
- p(x) = p(-x) for all x ∈ X.
Topology induced by a single F-seminorm
Theorem[10] — Let p be an F-seminorm on a vector space X. Then the map d : X × X → ℝ defined by d(x, y) := p(x - y) is a translation invariant pseudometric on X that defines a vector topology τ on X. If p is an F-norm then d is a metric. When X is endowed with this topology then p is a continuous map on X.
The balanced sets { x X : p(x) ≤ r }, as r ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. Similarly, the balanced sets { x X : p(x) < r }, as r 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 𝒮 is a non-empty collection of F-seminorms on a vector space X and for any finite subset ℱ ⊆ 𝒮 and any r > 0, let
- Uℱ, r := { x ∈ X : p(x) < r }.
The set { Uℱ, r : r > 0, ℱ ⊆ 𝒮, ℱ finite } forms a filter base on X that also forms a neighborhood basis at the origin for a vector topology on X denoted by τ𝒮.[11]
- Each Uℱ, r is a balanced and absorbing subset of X.[11]
- Uℱ, r/2 + Uℱ, r/2 ⊆ Uℱ, r.[11]
- τ𝒮 is the coarsest vector topology on X making each p ∈ 𝒮 continuous.[11]
- τ𝒮 is Hausdorff if and only if for every non-zero x ∈ X, there exists some p ∈ 𝒮 such that p(x) > 0.[11]
- If 𝒯 is the set of all continuous F-seminorms on (X, τ𝒮) then τ𝒮 = τ𝒯.[11]
- If 𝒯 is the set of all pointwise suprema of non-empty finite subsets of ℱ of 𝒮 then 𝒯 is a directed family of F-seminorms and τ𝒮 = τ𝒯.[11]
Fréchet combination
Suppose that p• = (pi)∞
i=1 is a family of non-negative subadditive functions on a vector space X.
Definition:[7] The Fréchet combination of p• is defined to be the real-valued map
- .
As an F-seminorm
Assume that p• = (pi)∞
i=1 is an increasing sequence of seminorms on X and let p be the Fréchet combination of p•.
Then p is an F-seminorm on X that induces the same locally convex topology as the family p• of seminorms.[13]
Since p• = (pi)∞
i=1 is increasing, a basis of open neighborhoods of the origin consists of all sets of the form { x ∈ X : pi(x) < r } as i ranges over all positive integers and r > 0 ranges over all positive real numbers.
The translation invariant pseudometric on X induced by this F-seminorm p is
(this metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations).[14]
As a paranorm
If each pi is a paranorm then so is p and moreover, p induces the same topology on X as the family p• of paranorms.[7] This is also true of the following paranorms on X:
Generalization
The Fréchet combination can be generalized by use of a bounded remetrization function.
- Definition:[15] A bounded remetrization function is a continuous non-negative non-decreasing map R : [0, ∞) → [0, ∞) that is subadditive (i.e. R (s + t) ≤ R (s) + R (t) for all s, t ≥ 0), has a bounded range, and satisfies R (s) = 0 if and only if s = 0.
Examples of bounded remetrization functions include arctan t, tanh t, t ↦ min { t, 1 }, and t ↦ t/1 + t.[15] If d is a pseudometric (resp. metric) on X and R} is a bounded remetrization function then R ∘ d is a bounded pseudometric (resp. bounded metric) on X that is uniformly equivalent to d.[15]
Suppose that p• = (pi)∞
i=1 is a family of non-negative F-seminorm on a vector space X, R} is a bounded remetrization function, and r• = (ri)∞
i=1 is a sequence of positive real numbers whose sum is finite.
Then
defines a bounded F-seminorm that is uniformly equivalent to the p•.[16] It has the property that for any net x• = (xi)a ∈ A in X, p (x•) → 0 if and only if pi (x•) → 0 for all i.[16] p is an F-norm if and only if the p• separate points on X.[16]
Characterizations
Of pseudometrizable TVS
If (X, τ) is a topological vector space (TVS) (where note in particular that τ is assumed to be a vector topology) then the following are equivalent:[10]
- X is pseudometrizable (i.e. the vector topology τ is induced by a pseudometric on X).
- X has a countable neighborhood base at the origin.
- The topology on X is induced by a translation-invariant pseudometric on X.
- The topology on X is induced by an F-seminorm.
- The topology on X is induced by a paranorm.
Of metrizable TVS
If (X, τ) is a TVS then the following are equivalent:
- X is metrizable.
- X is Hausdorff and pseudometrizable.
- X is Hausdorff and has a countable neighborhood base at the origin.[10][11]
- The topology on X is induced by a translation-invariant metric on X.[10]
- The topology on X is induced by an F-norm.[10][11]
- The topology on X is induced by a monotone F-norm.[11]
- The topology on X is induced by a total paranorm.
Of locally convex pseudometrizable TVS
If (X, τ) is TVS then the following are equivalent:[13]
- X is locally convex and pseudometrizable.
- X has a countable neighborhood base at the origin consisting of convex sets.
- The topology of X is induced by a countable family of (continuous) seminorms.
- The topology of X is induced by a countable increasing sequence of (continuous) seminorms (pi)∞
i=1 (increasing means that for all i, pi ≤ pi+1). - The topology of X is induced by an F-seminorm of the form:
i=1 are (continuous) seminorms on X.[17]
Quotients
Let M be a vector subspace of a topological vector space (X, τ).
- If X is a pseudometrizable TVS then so is X/M.[10]
- If X is a complete pseudometrizable TVS and M is a closed vector subspace of X then X/M is complete.[10]
- If X is metrizable TVS and M is a closed vector subspace of X then X/M is metrizable.[10]
- If p is an F-seminorm on X, then the map P : X/M → ℝ defined by
- P(x + M) := inf { p(x + m) : m ∈ M }
is an F-seminorm on X/M that induces the usual quotient topology on X/M.[10]
- If in addition p is an F-norm on X and if M is a closed vector subspace of X then P is an F-norm on X.[10]
Examples and sufficient conditions
- Every seminormed space (X, p) is pseudometrizable with a canonical pseudometric given by d(x, y) := p(x - y) for all x, y ∈ X.[18].
- If (X, d) is pseudometric TVS with a translation invariant pseudometric d, then p(x) := d(x, 0) defines a paranorm.[19]
- However, if d is a translation invariant pseudometric on the vector space X (without the addition condition that (X, d) is pseudometric TVS), then d need not be either an F-seminorm[20] nor a paranorm.
- If a TVS has a bounded neighborhood of 0 then it is pseudometrizable; the converse is in general false.[14]
- If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.[14]
- If a TVS X has a convex bounded neighborhood of the origin then it is seminormable ; if in addition X is Hausdorff then it is normable.[14]
If X is a Hausdorff locally convex TVS then the following are equivalent:
- X is normable.
- X has a bounded neighborhood of the origin.
- the strong dual of X is normable.[21]
- the strong dual of X is metrizable.[21]
If X is Hausdorff locally convex TVS then X with the strong topology, (X, b(X, X')), is metrizable if and only if there exists a countable set ℬ of bounded subsets of X such that every bounded subset of X is contained in some element of ℬ.[22]
Metrically bounded sets and bounded sets
Suppose that (X, d) is a pseudometric space and B ⊆ X. We say that B is metrically bounded or d-bounded if there exists a real number R > 0 such that d(x, y) ≤ R for all x, y ∈ B; the smallest such R is then called the diameter or d-diameter of B.[14] If B is bounded in a pseudometrizable TVS X then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.[14]
Properties of pseudometrizable TVS
Theorem[23] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.
Every metrizable locally convex TVS is a quasibarrelled space,[24] bornological space, and a Mackey space.
If X is a metrizable locally convex space, then the strong dual of X is bornological if and only if it is infrabarreled, if and only if it is barreled.[25]
If X is a complete pseudo-metrizable TVS and M is a closed vector subspace of X, then X/M is complete.[10]
The strong dual of a locally convex metrizable TVS is a webbed space.[26]
If (X, 𝜏) and (X, 𝜐) are complete metrizable TVSs and if 𝜐 is coarser than 𝜏 then 𝜏 = 𝜐.[27] This is no longer true if either one of these metrizable TVSs is not complete.[28]
Completeness
Recall that 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 X is a metrizable TVS and d is a metric that defines X's topology, then its possible that X is complete as a TVS (i.e. relative to its uniformity) but the metric d is not a complete metric (such metrics exist even for X = ℝ). Thus, if X is a TVS whose topology is induced by a pseudometric d, then the notion of completeness of X (as a TVS) and the notion of completeness of the pseudometric space (X, d) are not always equivalent. The next theorem gives a condition for when they are equivalent:
Theorem[29] — If X is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric d, then d is a complete pseudometric on X if and only if X is complete as a TVS.
Theorem[30] — If X is a TVS whose topology is induced by a paranorm p, then X is complete if and only if for every sequence (xi)∞
i=1 in X, if ∑∞
i=1 p(xi) < ∞ then ∑∞
i=1 xi converges in X.
Subsets and subsequences
- In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[31]
- If d is a translation invariant metric on a vector space X, then d(nx, 0) ≤ nd(x, 0) for all x ∈ X and every positive integer n.[32]
- If (xi)∞
i=1 is a null sequence (i.e. it converges to the origin) in a metrizable TVS then there exists a sequence (ri)∞
i=1 of positive real numbers diverging to ∞ such that (rixi)∞
i=1 → 0.[32]
Banach-Saks theorem[33] — If (xn)∞
n=1 is a sequence in a locally convex metrizable TVS (X, 𝜏) that converges weakly to some x ∈ X, then there exists a sequence y• = (yi)∞
i=1 in X such that y• → x in (X, 𝜏) and each yi is a convex combination of finitely many xn.
Mackey's countability condition[14] — Suppose that X is a locally convex metrizable TVS and that (Bi)∞
i=1 is a countable sequence of bounded subsets of X.
Then there exists a bounded subset B of X and a sequence (ri)∞
i=1 of positive real numbers such that Bi ⊆ ri B for all i.
Linear maps
- If X is a pseudometrizable TVS and A maps bounded subsets of X to bounded subsets of Y, then A is continuous.[14]
- Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.[34]
- Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[34]
If F : X → Y is a linear map between TVSs and X is metrizable then the following are equivalent:
- F is continuous;
- F is a (locally) bounded map (i.e. F maps (von Neumann) bounded subsets of X to bounded subsets of Y);[11]
- F is sequentially continuous;[11]
- the image under F of every null sequence in X is a bounded set;[11]
- Recall that a null sequence is a sequence that converges to the origin.
- F maps null sequences to null sequences;
Hahn-Banach extension property
Let X be a TVS. Say that a vector subspace M of X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X.[22] Say that X has the Hahn-Banach extension property (HBEP) if every vector subspace of X has the extension property.[22]
The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:
Theorem[22] (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.
If a vector space X 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.[22]
See also
- Asymmetric norm – Generalization of the concept of a norm
- Complete metric space
- Complete topological vector space
- Closed graph theorem
- F-space – Topological vector space with a complete translation-invariant metric
- Fréchet space
- Locally convex topological vector space – Type of topological vector space
- Metric space – Mathematical set defining distance
- Open mapping theorem (functional analysis)
- Pseudometric space
- Relation of norms and metrics
- Seminorm
- Sublinear function
- Topological vector space – Vector space with a notion of continuity
- Uniform space – Topological space with a notion of uniform properties
- Ursescu theorem
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