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In
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a metrizable (resp. pseudometrizable)
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(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 In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
metrizable TVS.


Pseudometrics and metrics

A pseudometric on a set X is a map d : X \times X \rarr \R satisfying the following properties:
  1. d(x, x) = 0 \text x \in X;
  2. Symmetry: d(x, y) = d(y, x) \text x, y \in X;
  3. Subadditivity In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
    : d(x, z) \leq d(x, y) + d(y, z) \text x, y, z \in X.
A pseudometric is called a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
if it satisfies:
  1. Identity of indiscernibles The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
    : for all x, y \in 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:
  1. Strong/Ultrametric triangle inequality: d(x, z) \leq \max \ \text x, y, z \in X.
Pseudometric space 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 In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
(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: B_r(z) := \ as z ranges over X and r > 0 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. :: If (X, d) is a pseudometric space and X is treated as a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, then unless indicated otherwise, it should be assumed that X is endowed with the topology induced by d. Pseudometrizable space A topological space (X, \tau) is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) d on X such that \tau is equal to the topology induced by d.


Pseudometrics and values on topological groups

An additive
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators. A topology \tau 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 (that is, if it makes X into a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
). Every
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(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 In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
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:
  1. Translation invariance In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equa ...
    : d(x + z, y + z) = d(x, y) \text x, y, z \in X;
  2. d(x, y) = d(x - y, 0) \text x, y \in X.


Value/G-seminorm

If X is a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
the a value or G-seminorm on X (the ''G'' stands for Group) is a real-valued map p : X \rarr \R with the following properties:
  1. Non-negative: p \geq 0.
  2. Subadditive: p(x + y) \leq p(x) + p(y) \text x, y \in X;
  3. p(0) = 0..
  4. Symmetric: p(-x) = p(x) \text x \in X.
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 values

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


Equivalence on topological groups


Pseudometrizable topological groups


An invariant pseudometric that doesn't induce a vector topology

Let X be a non-trivial (i.e. X \neq \) 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 \text x, y \in X such that x \neq y. The topology \tau that d induces on X is the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
, which makes (X, \tau) into a commutative topological group under addition but does form a vector topology on X because (X, \tau) is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on (X, \tau). 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 \mathcal of subsets of a vector space is called additive if for every N \in \mathcal, there exists some U \in \mathcal such that U + U \subseteq N. 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 In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
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 group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s. Assume that n_ = \left(n_1, \ldots, n_k\right) always denotes a finite sequence of non-negative integers and use the notation: \sum 2^ := 2^ + \cdots + 2^ \quad \text \quad \sum U_ := U_ + \cdots + U_. For any integers n \geq 0 and d > 2, U_n \supseteq U_ + U_ \supseteq U_ + U_ + U_ \supseteq U_ + U_ + \cdots + U_ + U_ + U_. From this it follows that if n_ = \left(n_1, \ldots, n_k\right) consists of distinct positive integers then \sum U_ \subseteq U_. It will now be shown by induction on k that if n_ = \left(n_1, \ldots, n_k\right) consists of non-negative integers such that \sum 2^ \leq 2^ for some integer M \geq 0 then \sum U_ \subseteq U_M. This is clearly true for k = 1 and k = 2 so assume that k > 2, which implies that all n_i are positive. If all n_i are distinct then this step is done, and otherwise pick distinct indices i < j such that n_i = n_j and construct m_ = \left(m_1, \ldots, m_\right) from n_ by replacing each n_i with n_i - 1 and deleting the j^ element of n_ (all other elements of n_ are transferred to m_ unchanged). Observe that \sum 2^ = \sum 2^ and \sum U_ \subseteq \sum U_ (because U_ + U_ \subseteq U_) so by appealing to the inductive hypothesis we conclude that \sum U_ \subseteq \sum U_ \subseteq U_M, as desired. It is clear that f(0) = 0 and that 0 \leq f \leq 1 so to prove that f is subadditive, it suffices to prove that f(x + y) \leq f(x) + f(y) when x, y \in X are such that f(x) + f(y) < 1, which implies that x, y \in U_0. This is an exercise. If all U_i are symmetric then x \in \sum U_ if and only if - x \in \sum U_ from which it follows that f(-x) \leq f(x) and f(-x) \geq f(x). If all U_i are balanced then the inequality f(s x) \leq f(x) for all unit scalars s such that , s, \leq 1 is proved similarly. Because f is a nonnegative subadditive function satisfying f(0) = 0, as described in the article on sublinear functionals, f is uniformly continuous on X if and only if f is continuous at the origin. If all U_i are neighborhoods of the origin then for any real r > 0, pick an integer M > 1 such that 2^ < r so that x \in U_M implies f(x) \leq 2^ < r. If the set of all U_i form basis of balanced neighborhoods of the origin then it may be shown that for any n > 1, there exists some 0 < r \leq 2^ such that f(x) < r implies x \in U_n. \blacksquare


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 \rarr \R on X that satisfies any of the following additional conditions, each of which begins with "for all sequences x_ = \left(x_i\right)_^ in X and all convergent sequences of scalars s_ = \left(s_i\right)_^":
  1. Continuity of multiplication: if s is a scalar and x \in X are such that p\left(x_i - x\right) \to 0 and s_ \to s, then p\left(s_i x_i - s x\right) \to 0.
  2. Both of the conditions: * if s_ \to 0 and if x \in X is such that p\left(x_i - x\right) \to 0 then p\left(s_i x_i\right) \to 0; * if p\left(x_\right) \to 0 then p\left(s x_i\right) \to 0 for every scalar s.
  3. Both of the conditions: * if p\left(x_\right) \to 0 and s_ \to s for some scalar s then p\left(s_i x_i\right) \to 0; * if s_ \to 0 then p\left(s_i x\right) \to 0 \text x \in X.
  4. Separate continuity: * if s_ \to s for some scalar s then p\left(s x_i - s x\right) \to 0 for every x \in X; * if s is a scalar, x \in X, and p\left(x_i - x\right) \to 0 then p\left(s x_i - s x\right) \to 0 .
A paranorm is called total if in addition it satisfies:


Properties of paranorms

If p is a paranorm on a vector space X then the map d : X \times X \rarr \R defined by d(x, y) := p(x - y) is a translation-invariant pseudometric on X that defines a on X. If p is a paranorm on a vector space X then:


Examples of paranorms


''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 \rarr \R with the following properties:
  1. Non-negative: p \geq 0.
  2. Subadditive: p(x + y) \leq p(x) + p(y) \text x, y \in X;
  3. Balanced: p(a x) \leq p(x) for all x \in X and all scalars a satisfying , a, \leq 1; * This condition guarantees that each set of the form \ or \ for some r \geq 0 is balanced.
  4. for every x \in X, p\left(\frac x\right) \to 0 as n \to \infty * The sequence \left(\frac\right)_^ can be replaced by any positive sequence converging to 0.
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(r x) < p(s x) for all non-zero x \in X and all real s and t such that s < t.


''F''-seminormed spaces

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. If (X, p) and (Z, q) are ''F''-seminormed spaces then a map f : X \to Z is called an isometric embedding if q(f(x) - f(y)) = p(x, y) \text x, y \in X. 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


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 X is a value on X. In particular, p(x) = 0, and p(x) = p(-x) for all x \in X.


Topology induced by a single ''F''-seminorm


Topology induced by a family of ''F''-seminorms

Suppose that \mathcal is a non-empty collection of ''F''-seminorms on a vector space X and for any finite subset \mathcal \subseteq \mathcal and any r > 0, let U_ := \bigcap_ \. The set \left\ forms a filter base on X that also forms a neighborhood basis at the origin for a vector topology on X denoted by \tau_. Each U_ is a balanced and absorbing subset of X. These sets satisfy U_ + U_ \subseteq U_.


Fréchet combination

Suppose that p_ = \left(p_i\right)_^ is a family of non-negative subadditive functions on a vector space X. The Fréchet combination of p_ is defined to be the real-valued map p(x) := \sum_^ \frac.


As an ''F''-seminorm

Assume that p_ = \left(p_i\right)_^ 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. Since p_ = \left(p_i\right)_^ is increasing, a basis of open neighborhoods of the origin consists of all sets of the form \left\ as i ranges over all positive integers and r > 0 ranges over all positive real numbers. The
translation invariant In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equa ...
pseudometric on X induced by this ''F''-seminorm p is d(x, y) = \sum^_ \frac \frac. 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 p_i is a paranorm then so is p and moreover, p induces the same topology on X as the family p_ of paranorms. This is also true of the following paranorms on X:


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 R : subadditive_In_mathematics,_subadditivity_is_a_property_of_a_function_that_states,_roughly,_that_evaluating_the_function_for_the_sum_of_two__elements_of_the_domain_always_returns_something_less_than_or_equal_to_the_sum_of_the_function's_values_at_each_element.__...
_(i.e._R(s_+_t)_\leq_R(s)_+_R(t)_for_all_s,_t_\geq_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_\mapsto_\min_\,_and_t_\mapsto_\frac._ If_d_is_a_pseudometric_(resp._metric)_on_X_and_R_is_a_bounded_remetrization_function_then_R_\circ_d_is_a_bounded_pseudometric_(resp._bounded_metric)_on_X_that_is_uniformly_equivalent_to_d._ Suppose_that_p__=_\left(p_i\right)_^_is_a_family_of_non-negative_''F''-seminorm_on_a_vector_space_X,_R}_is_a_bounded_remetrization_function,_and_r__=_\left(r_i\right)_^_is_a_sequence_of_positive_real_numbers_whose_sum_is_finite._ Then_ p(x)_:=_\sum_^_r_i_R\left(p_i(x)\right) defines_a_bounded_''F''-seminorm_that_is_uniformly_equivalent_to_the_p_._ It_has_the_property_that_for_any_net_x__=_\left(x_a\right)__in_X,_p\left(x_\right)_\to_0_if_and_only_if_p_i\left(x_\right)_\to_0_for_all_i._ p_is_an_''F''-norm_if_and_only_if_the_p__separate_points_on_X.


_Characterizations


_Of_(pseudo)metrics_induced_by_(semi)norms

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__for_all_scalars_s_and_all_x,_y_\in_X,_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_TVS

If_(X,_\tau)_is_a_topological_vector_space_ In_mathematics,_a_topological_vector_space_(also_called_a_linear_topological_space_and_commonly_abbreviated_TVS_or_t.v.s.)_is_one_of_the_basic_structures_investigated_in_functional_analysis. A_topological_vector_space_is_a_vector_space_that_is_als_...
_(TVS)_(where_note_in_particular_that_\tau_is_assumed_to_be_a_vector_topology)_then_the_following_are_equivalent:_
  1. X_is_pseudometrizable_(i.e._the_vector_topology_\tau_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_TVS

If_(X,_\tau)_is_a_TVS_then_the_following_are_equivalent:
  1. X_is_metrizable.
  2. X_is_Hausdorff_space.html" ;"title=", \infty) \to [0, \infty) that is
    subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
    (i.e. R(s + t) \leq R(s) + R(t) for all s, t \geq 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 \mapsto \min \, and t \mapsto \frac. If d is a pseudometric (resp. metric) on X and R is a bounded remetrization function then R \circ d is a bounded pseudometric (resp. bounded metric) on X that is uniformly equivalent to d. Suppose that p_ = \left(p_i\right)_^ is a family of non-negative ''F''-seminorm on a vector space X, R} is a bounded remetrization function, and r_ = \left(r_i\right)_^ is a sequence of positive real numbers whose sum is finite. Then p(x) := \sum_^ r_i R\left(p_i(x)\right) defines a bounded ''F''-seminorm that is uniformly equivalent to the p_. It has the property that for any net x_ = \left(x_a\right)_ in X, p\left(x_\right) \to 0 if and only if p_i\left(x_\right) \to 0 for all i. p is an ''F''-norm if and only if the p_ separate points on X.


    Characterizations


    Of (pseudo)metrics induced by (semi)norms

    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 for all scalars s and all x, y \in X, 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 TVS

    If (X, \tau) is a
    topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
    (TVS) (where note in particular that \tau is assumed to be a vector topology) then the following are equivalent:
    1. X is pseudometrizable (i.e. the vector topology \tau 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 TVS

    If (X, \tau) is a TVS then the following are equivalent:
    1. X is metrizable.
    2. X is Hausdorff space">Hausdorff and pseudometrizable.
    3. X is Hausdorff and has a countable neighborhood base at the origin.
    4. The topology on X is induced by a translation-invariant metric on X.
    5. The topology on X is induced by an ''F''-norm.
    6. The topology on X is induced by a monotone ''F''-norm.
    7. The topology on X is induced by a total paranorm.


    Of locally convex pseudometrizable TVS

    If (X, \tau) is TVS then the following are equivalent:
    1. X is
      locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
      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 \left(p_i\right)_^ (increasing means that for all i, p_i \geq p_.
    5. The topology of X is induced by an ''F''-seminorm of the form: p(x) = \sum_^ 2^ \operatorname p_n(x) where \left(p_i\right)_^ are (continuous) seminorms on X.


    Quotients

    Let M be a vector subspace of a topological vector space (X, \tau).
    • If X is a pseudometrizable TVS then so is X / M.
    • If X is a complete pseudometrizable TVS and M is a closed vector subspace of X then X / M is complete.
    • If X is metrizable TVS and M is a closed vector subspace of X then X / M is metrizable.
    • If p is an ''F''-seminorm on X, then the map P : X / M \to \R defined by P(x + M) := \inf_ \ is an ''F''-seminorm on X / M that induces the usual quotient topology on X / M. 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.


    Examples and sufficient conditions

    • Every
      seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
      (X, p) is pseudometrizable with a canonical pseudometric given by d(x, y) := p(x - y) for all x, y \in X..
    • If (X, d) is pseudometric ''TVS'' with a translation invariant pseudometric d, then p(x) := d(x, 0) defines a paranorm. However, if d is a translation invariant pseudometric on the vector space X (without the addition condition that (X, d) is ), then d need not be either an ''F''-seminorm nor a paranorm.
    • If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.
    • If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.
    • Suppose X is either a
      DF-space In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
      or an
      LM-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 ...
      . 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, \left(X, b\left(X, X^\right)\right), is metrizable if and only if there exists a countable set \mathcal of bounded subsets of X such that every bounded subset of X is contained in some element of \mathcal. The strong dual space X_b^ of a metrizable locally convex space (such as a Fréchet spaceGabriyelyan, S.S
    "On topological spaces and topological groups with certain local countable networks
    (2014)
    ) X is a
    DF-space In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
    . 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 X is a metrizable locally convex space then its strong dual X_b^ 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 Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
    bounded neighborhood of the origin. Moreover, a TVS is
    normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
    if and only if it is Hausdorff and seminormable. Every metrizable TVS on a finite-
    dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
    vector space is a normable
    locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
    complete TVS In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point x towards ...
    , being TVS-isomorphic to
    Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
    . Consequently, any metrizable TVS that is normable must be infinite dimensional. If M is a metrizable locally convex TVS that possess a
    countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
    fundamental system of bounded sets, then M is normable. If X is a Hausdorff
    locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
    then the following are equivalent:
    1. X is
      normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
      .
    2. X has a (von Neumann) bounded neighborhood of the origin.
    3. the strong dual space X^_b of X is normable.
    and if this locally convex space X is also metrizable, then the following may be appended to this list:
    1. the strong dual space of X is metrizable.
    2. the strong dual space of X is a Fréchet–Urysohn locally convex space.
    In particular, if a metrizable locally convex space X (such as a Fréchet space) is normable then its strong dual space X^_b is not a
    Fréchet–Urysohn space In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a spec ...
    and consequently, this
    complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
    Hausdorff locally convex space X^_b is also neither metrizable nor normable. Another consequence of this is that if X is a reflexive
    locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
    TVS whose strong dual X^_b is metrizable then X^_b is necessarily a reflexive Fréchet space, X is a
    DF-space In the field of functional analysis, DF-spaces, also written (''DF'')-spaces are locally convex topological vector space having a property that is shared by locally convex metrizable topological vector spaces. They play a considerable part in the ...
    , both X and X^_b are necessarily
    complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
    Hausdorff ultrabornological distinguished webbed spaces, and moreover, X^_b is normable if and only if X is normable if and only if X is Fréchet–Urysohn if and only if X is metrizable. In particular, such a space X is either a
    Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
    or else it is not even a Fréchet–Urysohn space.


    Metrically bounded sets and bounded sets

    Suppose that (X, d) is a pseudometric space and B \subseteq X. The set B is metrically bounded or d-bounded if there exists a real number R > 0 such that d(x, y) \leq R for all x, y \in B; the smallest such R is then called the diameter or d-diameter of B. 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 In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
    metrizable TVSs.


    Properties of pseudometrizable TVS

    • Every metrizable
      locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
      TVS is a
      quasibarrelled space In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied beca ...
      , bornological space, and a
      Mackey space In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space ''X'' such that the topology of ''X'' coincides with the Mackey topology τ(''X'',''X′''), the finest topology which still prese ...
      .
    • Every complete metrizable TVS is a
      barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a ...
      and a Baire space (and hence non-meager). However, there exist metrizable Baire spaces that are not
      complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
      .
    • If X is a metrizable locally convex space, then the strong dual of X is bornological if and only if it is barreled, if and only if it is infrabarreled.
    • If X is a complete pseudometrizable TVS and M is a closed vector subspace of X, then X / M is complete.
    • The
      strong dual In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
      of a locally convex metrizable TVS is a webbed space.
    • If (X, \tau) and (X, \nu) are complete metrizable TVSs (i.e.
      F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
      s) and if \nu is coarser than \tau then \tau = \nu; this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete. Said differently, if (X, \tau) and (X, \nu) are both
      F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
      s but with different topologies, then neither one of \tau and \nu contains the other as a subset. One particular consequence of this is, for example, that if (X, p) is a
      Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
      and (X, q) is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of (X, p) (i.e. if p \leq C q or if q \leq C p for some constant C > 0), then the only way that (X, q) can be a Banach space (i.e. also be complete) is if these two norms p and q are equivalent; if they are not equivalent, then (X, q) can not be a Banach space. As another consequence, if (X, p) is a Banach space and (X, \nu) is a Fréchet space, then the map p : (X, \nu) \to \R is continuous if and only if the Fréchet space (X, \nu) the TVS (X, p) (here, the Banach space (X, p) is being considered as a TVS, which means that its norm is " forgetten" but its topology is remembered).
    • A metrizable
      locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
      space is
      normable In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is z ...
      if and only if its strong dual space is a Fréchet–Urysohn locally convex space.
    • Any product of complete metrizable TVSs is a Baire space.
    • A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension 0.
    • A product of pseudometrizable TVSs is pseudometrizable if and only if it all but at most countably many of these TVSs have the trivial topology.
    • Every complete metrizable TVS is a
      barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a ...
      and a Baire space (and thus non-meager).
    • The dimension of a complete metrizable TVS is either finite or uncountable.


    Completeness

    Every
    topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
    (and more generally, a
    topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
    ) 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 a
    complete metric In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in . Intuitively, a space is complete if ther ...
    (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: If M is a closed vector subspace of a complete pseudometrizable TVS X, then the quotient space X / M is complete. If M is a vector subspace of a metrizable TVS X and if the quotient space X / M is complete then so is X. If X is not complete then M := X, but not complete, vector subspace of X. A Baire separable
    topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
    is metrizable if and only if it is cosmic.Gabriyelyan, S.S
    "On topological spaces and topological groups with certain local countable networks
    (2014)


    Subsets and subsequences

    • 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 \subseteq \operatorname_C R.
    • Every totally bounded subset of a locally convex metrizable TVS X is contained in the closed
      convex balanced hull In mathematics, a subset ''C'' of a Real number, real or Complex number, complex vector space is said to be absolutely convex or disked if it is Convex set, convex and Balanced set, balanced (some people use the term "circled" instead of "balanced") ...
      of some sequence in X that converges to 0.
    • In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.
    • If d is a translation invariant metric on a vector space X, then d(n x, 0) \leq n d(x, 0) for all x \in X and every positive integer n.
    • If \left(x_i\right)_^ is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence \left(r_i\right)_^ of positive real numbers diverging to \infty such that \left(r_i x_i\right)_^ \to 0.
    • A subset of a complete metric space is closed if and only if it is complete. 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 In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
      TVS then for every bounded subset B of X, there exists a bounded disk D in X such that B \subseteq X_D, and both X and the
      auxiliary normed space In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk D is bounded: in this case, the au ...
      X_D induce the same subspace topology on B.


    Linear maps

    If X is a pseudometrizable TVS and A maps bounded subsets of X to bounded subsets of Y, then A 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 F : X \to Y 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 (that is, F maps (von Neumann) bounded subsets of X to bounded subsets of Y);
    3. F is sequentially continuous;
    4. the image under F of every null sequence in X is a bounded set where by definition, a is a sequence that converges to the origin.
    5. F maps null sequences to null sequences;
    Open and almost open maps :Theorem: If X is a complete pseudometrizable TVS, Y is a Hausdorff TVS, and T : X \to Y is a closed and almost open linear surjection, then T is an open map. :Theorem: If T : X \to Y is a surjective linear operator from a
    locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
    space X onto a
    barrelled space In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a ...
    Y (e.g. every complete pseudometrizable space is barrelled) then T is almost open. :Theorem: If T : X \to Y is a surjective linear operator from a TVS X onto a Baire space Y then T is almost open. :Theorem: Suppose T : X \to Y 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 : X \to Y is a surjective open map and Y is a
    complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
    metrizable space.


    Hahn-Banach extension property

    A vector subspace M of a TVS X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X. Say that a TVS X has the Hahn-Banach extension property (HBEP) if every vector subspace of X 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 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.


    See also

    * * * * * * * * * * * * * * * *


    Notes


    References


    Bibliography

    * * * * * * * * * * * * * * * * * * {{Topological vector spaces Metric spaces Topological vector spaces