Metrizable topological vector space

  (Redirected from Paranorm)

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). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Pseudometrics and metricsEdit

A pseudometric on a set X is a map d : X × X → ℝ satisfying the following properties:

  1. d(x, x) = 0 for all xX;
  2. Symmetry: d(x, y) = d(y, x) for all x, yX;
  3. Subadditivity: d(x, z) ≤ d(x, y) + d(y, z) for all x, y, zX.

A pseudometric is called a metric if it satisfies:

  1. Identity of indiscernibles: for all x, yX, 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:

  1. Strong/Ultrametric triangle inequality: for all x, yX, 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 pseudometricEdit

If d is a pseudometric on a set X then collection of open balls:

Br(z) := { xX : 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 groupsEdit

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 pseudometricsEdit

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:

  1. Translation invariance: d(x + z, y + z) = d(x, y) for all x, y, zX;
  2. d(x, y) = d(x - y, 0) for all x, yX.

Value/G-seminormEdit

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]

  1. Non-negative: p ≥ 0.
  2. Subadditive: p(x+y) ≤ p(x) + p(y) for all x, yX;
  3. p(0) = 0.
  4. Symmetric: p(-x) = p(x) for all xX.

where we call a G-seminorm a G-norm if it satisfies the additional condition:

  1. Total/Positive definite: If p(x) = 0 then x = 0.

Properties of valuesEdit

If p is a value on a vector space X then:

  • |p(x) - p(y)| ≤ p(x - y) for all x, yX.[3]
  • p(nx) ≤ np(x) and 1/np(x) ≤ p(x/n)for all xX and positive integers n.[4]
  • The set { xX : p(x) = 0 } is an additive subgroup of X.[3]

Equivalence on topological groupsEdit

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). Conversely, if p is a value on X then the map d(x, y) := p(x - y) is a translation-invariant pseudometric on X and the value associated with d is just p.

Pseudometrizable topological groupsEdit

Theorem[2] — If (X, τ) is an additive commutative topological group then the following are equivalent:

  1. τ is induced by a pseudometric; (i.e. (X, τ) is pseudometrizable);
  2. τ is induced by a translation-invariant pseudometric;
  3. 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 topologyEdit

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, yX such that xy. 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.

Additive sequencesEdit

Definition:[5] A collection 𝒩 of subsets of a vector space is called additive if for every N ∈ 𝒩, there exists some U ∈ 𝒩 such that U + UN.

Continuity of addition at 0[5] — If (X, +) is a group (as all vector spaces are), τ is a topology on X, and X × X is endowed with the product topology, then the addition map X × XX (i.e. the map (x, y) ↦ x + y) is continuous at the origin of X × X if and only if the set of neighborhoods of the origin in (X, τ) 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.

Theorem — Let U = (Ui)
i=0
be a collection of subsets of a vector space such that 0 ∈ Ui and Ui+1 + Ui+1Ui for all i ≥ 0. For all uU0, let

𝕊(u) := { n = (n1, ⋅⋅⋅, nk) : k ≥ 1, ni ≥ 0 for all i, and uUn1 + ⋅⋅⋅ + Unk}.

Define f : X → [0, 1] by f (x) = 1 if xU0 and otherwise let

f (x) := inf { 2- n1 + ⋅⋅⋅ + 2- nk : n = (n1, ⋅⋅⋅, nk) ∈ 𝕊(x)  }.

Then f is subadditive (i.e. f (x + y) ≤ f (x) + f (y) for all x, yX) and f = 0 on i ≥ 0 Ui, so in particular f (0) = 0. If all Ui are symmetric sets then f (- x) = f (x) and if all Ui are balanced then f (s x) ≤ f (x) for all scalars s such that |s| ≤ 1 and all xX. If X is a topological vector space and if all Ui are neighborhoods of the origin then f is continuous, where if in addition X is Hausdorff and U forms a basis of balanced neighborhoods of the origin in X then d(x, y) := f (x - y) is a metric defining the vector topology on X.

Proof

We also assume that n = (n1, ⋅⋅⋅, nk) always denotes a finite sequence of non-negative integers and we will use the notation:

2- n  :=  2- n1 + ⋅⋅⋅ + 2- nk    and    Un  :=  Un1 + ⋅⋅⋅ + Unk.

Observe that for any integers n ≥ 0 and d > 2,

Un    Un+1 + Un+1    Un+1 + Un+2 + Un+2    Un+1 + Un+2 +  ⋅⋅⋅  + Un+d + Un+d+1 + Un+d+1.

From this it follows that if n = (n1, ⋅⋅⋅, nk) consists of distinct positive integers then UnU-1 + min (n).

We show by induction on k that if n = (n1, ⋅⋅⋅, nk) consists of non-negative integers such that 2- n ≤ 2- M for some integer M ≥ 0 then UnUM. This is clearly true for k = 1 and k = 2 so assume that k > 2, which implies that all ni are positive. If all ni are distinct then we're done, otherwise pick distinct indices i < j such that ni = nj and construct m = (m1, ..., mk-1) from n by replacing ni with ni - 1 and deleting the jth element of n (all other elements of n are transferred to m unchanged). Observe that 2- n = 2- m and Un Um (since Uni + UnjUni - 1) so by appealing to the inductive hypothesis we conclude that Un UmUM, as desired.

It is clear that f (0) = 0 and that 0 ≤ f ≤ 1 so to prove that f is subadditive, it suffices to prove that f (x + y) ≤ f (x) + f (y) when x, yX are such that f (x) + f (y) < 1, which implies that x, yU0. This is an exercise. If all Ui are symmetric then x Un if and only if - x Un from which it follows that f (-x) ≤ f (x) and f (-x) ≥ f (x). If all Ui are balanced then the inequality f (s x) ≤ f (x) for all unit scalars s is proved similarly. Since f is a nonnegative subadditive function satisfying f (0) = 0, f is uniformly continuous on X if and only if f is continuous at 0. If all Ui are neighborhoods of the origin then for any real r > 0, pick an integer M > 1 such that 2- M < r so that xUM implies f (x) ≤ 2- M < r. If all Ui form basis of balanced neighborhoods of the origin then one may show that for any n > 0, there exists some 0 < r ≤ 2- n such that f (x) < r implies xUn. ∎

ParanormsEdit

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
":[6]

  1. Continuity of multiplication: if s is a scalar and xX are such that p(xi - x) → 0 and ss, then p(si xi - sx) → 0.
  2. Both of the conditions:
    • if s → 0 and if xX 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.
  3. Both of the conditions:
    • if p(xi) → 0 and ss for some scalar s then p(si xi) → 0;
    • if s → 0 then p(si x) → 0 for all xX.
  4. Separate continuity:[7]
    • if ss for some scalar s then p(si x - sx) → 0 for every xX;
    • if s is a scalar, xX, and p(xi - x) → 0 then p(s xi - sx) → 0.

A paranorm is called total if in addition it satisfies:

  • Total/Positive definite: p(x) = 0 implies x = 0.

Properties of paranormsEdit

  • 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.[8]

If p is a paranorm on a vector space X then:

  • the set { xX : p(x) = 0 } is a vector subspace of X.[8]
  • p(x + n) = p(x) for all x, nX with p(n) = 0.[8]
  • If a paranorm p satisfies p(sx) ≤ |s| p(x) for all xX and scalars s, then p is absolutely homogeneity (i.e. equality holds)[8] and thus p is a seminorm.

Examples of paranormsEdit

  • 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.[8]
  • If p is a paranorm on X then so is the map q(x) := p(x)/[1 + p(x)].[8]
  • Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
  • Every seminorm is a paranorm.[8]
  • The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).[9]
  • The sum of two paranorms is a paranorm.[8]
  • If p and q are paranorms on X then so is (pq)(x) := inf { p(y) + q(z) : x = y + z with y, zX}. Moreover, pqp and pqq. This makes the set of paranorms on X into a conditionally complete lattice.[8]
  • 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.[8]
  • If x = (xi)iI is a Hamel basis on a vector space X then the real-valued map that sends x = iI sixiX (where all but finitely many of the scalars si are 0) to iI |si| is a paranorm on X, which satisfies p(sx) = |s| p(x) for all xX and scalars s.[8]
  • The function p(x) := |sin(πx)| + min {2, |x| } is a paranorm on that is not balanced but nevertheless equivalent to the usual norm on R. Note that the function x ↦ |sin(πx)| is subadditive.[10]
  • Let X be a complex vector space and let X denote X considered as a vctor space over . Any paranorm on X is also a paranorm on X.[9]

F-seminormsEdit

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:[11]

  1. Non-negative: p ≥ 0.
  2. Subadditive: p(x+y) ≤ p(x) + p(y) for all x, yX;
  3. Balanced: p(ax) ≤ p(x) for all xX and all scalars a satisfying |a| ≤ 1 ;
    • This condition guarantees that each set of the form { xX : p(x) ≤ r } or { xX : p(x) < r } for some r ≥ 0 is balanced.
  4. for every xX, p(1/n x) → 0 as n → ∞
    • The sequence (1/n)
      n=1
      can be replaced by any positive sequence converging to 0.[12]

An F-seminorm is called an F-norm if in addition it satisfies:

  1. Total/Positive definite: p(x) = 0 implies x = 0.

An F-seminorm is called monotone if it satisfies:

  1. Monotone: p(rx) < p(sx) for all non-zero xX and all real s and t such that s < t.[12]

F-seminormed spacesEdit

Definition:[12] 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:[12] If (X, p) and (Z, q) are F-seminormed spaces then a map f : XZ is called an isometric embedding if q(f (x) - f (y)) = p(x - y) for all x, yX.

Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.[12]

Examples of F-seminormsEdit

  • 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.[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 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.[10]
    • In particular, every seminorm is an F-seminorm.
  • For any 0 < p < 1, the map f on n defined by [f(x1, ..., xn)]p := |x1|p + ⋅⋅⋅ + |xn|p is an F-norm that is not a norm.
  • If L : XY is a linear map and if q is an F-seminorm on Y, then qL is an F-seminorm on X.[12]
  • Let X be a complex vector space and let X denote X considered as a vctor space over . Any F-seminorm on X is also an F-seminorm on X.[9]

Properties of F-seminormsEdit

  • Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.[7]
  • Every F-seminorm on a vector space X is a value on X. In particular,
    • p(0) = 0;
    • p(x) = p(-x) for all xX.

Topology induced by a single F-seminormEdit

Theorem[11] — 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-seminormsEdit

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 := p ∈ ℱ { xX : 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.[12]

  • Each Uℱ, r is a balanced and absorbing subset of X.[12]
  • Uℱ, r/2 + Uℱ, r/2Uℱ, r.[12]
  • τ𝒮 is the coarsest vector topology on X making each p ∈ 𝒮 continuous.[12]
  • τ𝒮 is Hausdorff if and only if for every non-zero xX, there exists some p ∈ 𝒮 such that p(x) > 0.[12]
  • If 𝒯 is the set of all continuous F-seminorms on (X, τ𝒮) then τ𝒮 = τ𝒯.[12]
  • If 𝒯 is the set of all pointwise suprema of non-empty finite subsets of of 𝒮 then 𝒯 is a directed family of F-seminorms and τ𝒮 = τ𝒯.[12]

Fréchet combinationEdit

Suppose that p = (pi)
i=1
is a family of non-negative subadditive functions on a vector space X.

Definition:[8] The Fréchet combination of p is defined to be the real-valued map

 .

As an F-seminormEdit

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 { xX : 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 paranormEdit

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.[8] This is also true of the following paranorms on X:

  • q(x) := inf { n
    i=1
    pi(x) + 1/n : n > 0 is an integer 
    }.[8]
  • r(x) :=
    n=1
    min { 1/2n, pn(x) 
    }.[8]

GeneralizationEdit

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 tt/1 + t.[15] If d is a pseudometric (resp. metric) on X and R} is a bounded remetrization function then Rd 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)aA 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]

CharacterizationsEdit

Of (pseudo)metrics induced by (semi)normsEdit

A pseudometric (resp. metric) d is induced by a seminorm (resp. norm) on a vector space X if and only if d is translation invariant and absolutely homogeneous, which means that d(sx, sy) = |s| d(x, y) for all scalars s and all x, yX, in which case the function defined by p(x) := d(x, 0) is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by p is equal to d.

Of pseudometrizable TVSEdit

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:[11]

  1. X is pseudometrizable (i.e. the vector topology τ is induced by a pseudometric on X).
  2. X has a countable neighborhood base at the origin.
  3. The topology on X is induced by a translation-invariant pseudometric on X.
  4. The topology on X is induced by an F-seminorm.
  5. The topology on X is induced by a paranorm.

Of metrizable TVSEdit

If (X, τ) is a TVS then the following are equivalent:

  1. X is metrizable.
  2. X is Hausdorff and pseudometrizable.
  3. X is Hausdorff and has a countable neighborhood base at the origin.[11][12]
  4. The topology on X is induced by a translation-invariant metric on X.[11]
  5. The topology on X is induced by an F-norm.[11][12]
  6. The topology on X is induced by a monotone F-norm.[12]
  7. The topology on X is induced by a total paranorm.

Birkhoff–Kakutani theorem — If (X, τ) is a topological vector space then the following three conditions are equivalent:[17][note 1]

  1. The origin { 0 } is closed in X, and there is a countable basis of neighborhoods for 0 in X.
  2. (X, τ) is metrizable (as a topological space).
  3. There is a translation-invariant metric on X that induces on X the topology τ, which is the given topology on X.

By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.

Of locally convex pseudometrizable TVSEdit

If (X, τ) is TVS then the following are equivalent:[13]

  1. X is locally convex and pseudometrizable.
  2. X has a countable neighborhood base at the origin consisting of convex sets.
  3. The topology of X is induced by a countable family of (continuous) seminorms.
  4. The topology of X is induced by a countable increasing sequence of (continuous) seminorms (pi)
    i=1
    (increasing means that for all i, pipi+1).
  5. The topology of X is induced by an F-seminorm of the form:
     
    where (pi)
    i=1
    are (continuous) seminorms on X.[18]

QuotientsEdit

Let M be a vector subspace of a topological vector space (X, τ).

  • If X is a pseudometrizable TVS then so is X/M.[11]
  • If X is a complete pseudometrizable TVS and M is a closed vector subspace of X then X/M is complete.[11]
  • If X is metrizable TVS and M is a closed vector subspace of X then X/M is metrizable.[11]
  • If p is an F-seminorm on X, then the map P : X/M → ℝ defined by
    P(x + M) := inf { p(x + m) : mM}
    is an F-seminorm on X/M that induces the usual quotient topology on X/M.[11]
    • 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.[11]

Examples and sufficient conditionsEdit

  • Every seminormed space (X, p) is pseudometrizable with a canonical pseudometric given by d(x, y) := p(x - y) for all x, yX.[19].
  • If (X, d) is pseudometric TVS with a translation invariant pseudometric d, then p(x) := d(x, 0) defines a paranorm.[20]
    • 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[21] 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]
  • Suppose X is either a DF-space or an LM-space. If X is a sequential space then it is either metrizable or else a Montel DF-space.

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 of bounded subsets of X such that every bounded subset of X is contained in some element of .[22]

NormabilityEdit

If X is a Hausdorff locally convex TVS then the following are equivalent:

  1. X is normable.
  2. X has a bounded neighborhood of the origin.
  3. the strong dual   of X is normable.[23]
  4. the strong dual   of X is metrizable.[23]

Moreover,

Metrically bounded sets and bounded setsEdit

Suppose that (X, d) is a pseudometric space and BX. 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, yB; 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 TVSEdit

Theorem[25] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.

CompletenessEdit

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[34] — 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[35][36] (Klee) — Let d be any[note 2] metric on a vector space X such that the topology 𝜏 induced by d on X makes (X, 𝜏) into a topological vector space. If (X, d) is a complete metric space then (X, 𝜏) is a complete-TVS.

Theorem[37] — 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.

  • A Baire separable topological group is metrizable if and only if it is cosmic.[32]
  • If M is a closed vector subspace of a complete pseudometrizable TVS X, then the quotient space XM is complete.[38]
  • Suppose M is a complete vector subspace of a metrizable TVS X. If the quotient space XM is complete then so is X.[38]
    • Note that if X is not complete then M := X is a closed, but not complete, vector subspace of X.

Subsets and subsequencesEdit

  • Let M be a separable locally convex metrizable topological vector space and let C be its completion. If S is a bounded subset of C then there exists a bounded subset R of X such that S ⊆ clC R.[39]
  • Every totally bounded subset of a locally convex metrizable TVS X is contained in the closed [[Absolutely convex set}convex balanced hull]] of some sequence in X that converges to 0.
  • In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[40]
  • If d is a translation invariant metric on a vector space X, then d(nx, 0) ≤ nd(x, 0) for all xX and every positive integer n.[41]
  • 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
    .[41]
  • A subset of a complete metric space is closed if and only if it is complete.
    • Note that if a space X is not complete, then X is a closed subset of X that is not complete.
  • If X is a metrizable locally convex TVS then for every bounded subset B of X, there exists a bounded disk D in X such that BXD, and both X and the auxiliary normed space XD induce the same subspace topology on B.[42]

Banach-Saks theorem[43] — If (xn)
n=1
is a sequence in a locally convex metrizable TVS (X, 𝜏) that converges weakly to some xX, then there exists a sequence y = (yi)
i=1
in X such that yx 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 Biri B for all i.

Linear mapsEdit

  • 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.[44]
    • Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[44]

If F : XY is a linear map between TVSs and X is metrizable then the following are equivalent:

  1. F is continuous;
  2. F is a (locally) bounded map (i.e. F maps (von Neumann) bounded subsets of X to bounded subsets of Y);[12]
  3. F is sequentially continuous;[12]
  4. the image under F of every null sequence in X is a bounded set;[12]
    • Recall that a null sequence is a sequence that converges to the origin.
  5. F maps null sequences to null sequences;
Open and almost open maps
Theorem:[45] If X is a complete pseudometrizable TVS, Y is a Hausdorff TVS, and T : XY is a closed and almost open linear surjection, then T is an open map.
Theorem:[45] If T : XY is a surjective linear operator from a locally convex space X onto a barrelled space Y then T is almost open. (note that every complete pseudometrizable space is barrelled)
Theorem:[45] If T : XY is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open.
Theorem:[45] Suppose T : XY is a continuous linear operator from a complete pseudometrizable TVS X into a Hausdorff TVS Y. If the image of T is non-meager in Y then T : XY is a surjective open map and Y is a complete metrizable space.

Hahn-Banach extension propertyEdit

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 alsoEdit

NotesEdit

  1. ^ In fact, this is true for topological group, for the proof doesn't use the scalar multiplications.
  2. ^ Not assumed to be translation-invariant.

ReferencesEdit

  1. ^ Narici & Beckenstein 2011, pp. 1-18.
  2. ^ a b c Narici & Beckenstein 2011, pp. 37-40.
  3. ^ a b Swartz 1992, p. 15.
  4. ^ Wilansky 2013, p. 17.
  5. ^ a b Wilansky 2013, pp. 40-47.
  6. ^ Wilansky 2013, p. 15.
  7. ^ a b Schechter 1996, pp. 689-691.
  8. ^ a b c d e f g h i j k l m n o Wilansky 2013, pp. 15-18.
  9. ^ a b c d Schechter 1996, p. 692.
  10. ^ a b Schechter 1996, p. 691.
  11. ^ a b c d e f g h i j k l Narici & Beckenstein 2011, pp. 91-95.
  12. ^ a b c d e f g h i j k l m n o p q r s t Jarchow 1981, pp. 38-42.
  13. ^ a b Narici & Beckenstein 2011, p. 123.
  14. ^ a b c d e f g h Narici & Beckenstein 2011, pp. 156-175.
  15. ^ a b c Schechter 1996, p. 487.
  16. ^ a b c Schechter 1996, pp. 692-693.
  17. ^ Köthe 1983, section 15.11
  18. ^ Schechter 1996, p. 706.
  19. ^ Narici & Beckenstein 2011, pp. 115-154.
  20. ^ Wilansky 2013, pp. 15-16.
  21. ^ Schaefer & Wolff 1999, pp. 91-92.
  22. ^ a b c d e Narici & Beckenstein 2011, pp. 225-273.
  23. ^ a b Trèves 2006, p. 201.
  24. ^ Schaefer & Wolff 1999, pp. 68-72.
  25. ^ Wilansky 2013, p. 57.
  26. ^ Jarchow 1981, p. 222.
  27. ^ a b c d Narici & Beckenstein 2011, pp. 371-423.
  28. ^ Schaefer & Wolff 1999, p. 153.
  29. ^ Narici & Beckenstein 2011, pp. 459-483.
  30. ^ Köthe 1969, p. 168.
  31. ^ Wilansky 2013, p. 59.
  32. ^ a b Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
  33. ^ a b Schaefer & Wolff 1999, pp. 12-35.
  34. ^ Narici & Beckenstein 2011, pp. 47-50.
  35. ^ Schaefer & Wolff 1999, p. 35.
  36. ^ Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4.
  37. ^ Wilansky 2013, pp. 56-57.
  38. ^ a b Narici & Beckenstein 2011, pp. 47-66.
  39. ^ Schaefer & Wolff 1999, pp. 190-202.
  40. ^ Narici & Beckenstein 2011, pp. 172-173.
  41. ^ a b Rudin 1991, p. 22.
  42. ^ Narici & Beckenstein 2011, pp. 441-457.
  43. ^ Rudin 1991, p. 67.
  44. ^ a b Narici & Beckenstein 2011, p. 125.
  45. ^ a b c d Narici & Beckenstein 2011, pp. 466-468.

BibliographyEdit