Metrizable topological vector space

Summary

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Pseudometrics and metrics edit

A pseudometric on a set   is a map   satisfying the following properties:

  1.  ;
  2. Symmetry:  ;
  3. Subadditivity:  

A pseudometric is called a metric if it satisfies:

  1. Identity of indiscernibles: for all   if   then  

Ultrapseudometric

A pseudometric   on   is called a ultrapseudometric or a strong pseudometric if it satisfies:

  1. Strong/Ultrametric triangle inequality:  

Pseudometric space

A pseudometric space is a pair   consisting of a set   and a pseudometric   on   such that  's topology is identical to the topology on   induced by   We call a pseudometric space   a metric space (resp. ultrapseudometric space) when   is a metric (resp. ultrapseudometric).

Topology induced by a pseudometric edit

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

 
as   ranges over   and   ranges over the positive real numbers, forms a basis for a topology on   that is called the  -topology or the pseudometric topology on   induced by  
Convention: If   is a pseudometric space and   is treated as a topological space, then unless indicated otherwise, it should be assumed that   is endowed with the topology induced by  

Pseudometrizable space

A topological space   is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric)   on   such that   is equal to the topology induced by  [1]

Pseudometrics and values on topological groups edit

An additive topological group is an additive group endowed with a topology, called a group topology, under which addition and negation become continuous operators.

A topology   on a real or complex vector space   is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes   into a topological vector space).

Every topological vector space (TVS)   is an additive commutative topological group but not all group topologies on   are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space   may fail to make scalar multiplication continuous. For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

Translation invariant pseudometrics edit

If   is an additive group then we say that a pseudometric   on   is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

  1. Translation invariance:  ;
  2.  

Value/G-seminorm edit

If   is a topological group the a value or G-seminorm on   (the G stands for Group) is a real-valued map   with the following properties:[2]

  1. Non-negative:  
  2. Subadditive:  ;
  3.  
  4. Symmetric:  

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

  1. Total/Positive definite: If   then  

Properties of values edit

If   is a value on a vector space   then:

  •  [3]
  •   and   for all   and positive integers  [4]
  • The set   is an additive subgroup of  [3]

Equivalence on topological groups edit

Theorem[2] — Suppose that   is an additive commutative group. If   is a translation invariant pseudometric on   then the map   is a value on   called the value associated with  , and moreover,   generates a group topology on   (i.e. the  -topology on   makes   into a topological group). Conversely, if   is a value on   then the map   is a translation-invariant pseudometric on   and the value associated with   is just  

Pseudometrizable topological groups edit

Theorem[2] — If   is an additive commutative topological group then the following are equivalent:

  1.   is induced by a pseudometric; (i.e.   is pseudometrizable);
  2.   is induced by a translation-invariant pseudometric;
  3. the identity element in   has a countable neighborhood basis.

If   is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric." A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable.

An invariant pseudometric that doesn't induce a vector topology edit

Let   be a non-trivial (i.e.  ) real or complex vector space and let   be the translation-invariant trivial metric on   defined by   and   such that   The topology   that   induces on   is the discrete topology, which makes   into a commutative topological group under addition but does not form a vector topology on   because   is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on  

This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F-seminorms.

Additive sequences edit

A collection   of subsets of a vector space is called additive[5] if for every   there exists some   such that  

Continuity of addition at 0 — If   is a group (as all vector spaces are),   is a topology on   and   is endowed with the product topology, then the addition map   (i.e. the map  ) is continuous at the origin of   if and only if the set of neighborhoods of the origin in   is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."[5]

All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that a Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups.

Theorem — Let   be a collection of subsets of a vector space such that   and   for all   For all   let

 

Define   by   if   and otherwise let

 

Then   is subadditive (meaning  ) and   on   so in particular   If all   are symmetric sets then   and if all   are balanced then   for all scalars   such that   and all   If   is a topological vector space and if all   are neighborhoods of the origin then   is continuous, where if in addition   is Hausdorff and   forms a basis of balanced neighborhoods of the origin in   then   is a metric defining the vector topology on  

Proof

Assume that   always denotes a finite sequence of non-negative integers and use the notation:

 

For any integers   and  

 

From this it follows that if   consists of distinct positive integers then  

It will now be shown by induction on   that if   consists of non-negative integers such that   for some integer   then   This is clearly true for   and   so assume that   which implies that all   are positive. If all   are distinct then this step is done, and otherwise pick distinct indices   such that   and construct   from   by replacing each   with   and deleting the   element of   (all other elements of   are transferred to   unchanged). Observe that   and   (because  ) so by appealing to the inductive hypothesis we conclude that   as desired.

It is clear that   and that   so to prove that   is subadditive, it suffices to prove that   when   are such that   which implies that   This is an exercise. If all   are symmetric then   if and only if   from which it follows that   and   If all   are balanced then the inequality   for all unit scalars   such that   is proved similarly. Because   is a nonnegative subadditive function satisfying   as described in the article on sublinear functionals,   is uniformly continuous on   if and only if   is continuous at the origin. If all   are neighborhoods of the origin then for any real   pick an integer   such that   so that   implies   If the set of all   form basis of balanced neighborhoods of the origin then it may be shown that for any   there exists some   such that   implies    

Paranorms edit

If   is a vector space over the real or complex numbers then a paranorm on   is a G-seminorm (defined above)   on   that satisfies any of the following additional conditions, each of which begins with "for all sequences   in   and all convergent sequences of scalars  ":[6]

  1. Continuity of multiplication: if   is a scalar and   are such that   and   then  
  2. Both of the conditions:
    • if   and if   is such that   then  ;
    • if   then   for every scalar  
  3. Both of the conditions:
    • if   and   for some scalar   then  ;
    • if   then  
  4. Separate continuity:[7]
    • if   for some scalar   then   for every  ;
    • if   is a scalar,   and   then   .

A paranorm is called total if in addition it satisfies:

  • Total/Positive definite:   implies  

Properties of paranorms edit

If   is a paranorm on a vector space   then the map   defined by   is a translation-invariant pseudometric on   that defines a vector topology on  [8]

If   is a paranorm on a vector space   then:

  • the set   is a vector subspace of  [8]
  •   with  [8]
  • If a paranorm   satisfies   and scalars   then   is absolutely homogeneity (i.e. equality holds)[8] and thus   is a seminorm.

Examples of paranorms edit

  • If   is a translation-invariant pseudometric on a vector space   that induces a vector topology   on   (i.e.   is a TVS) then the map   defines a continuous paranorm on  ; moreover, the topology that this paranorm   defines in   is  [8]
  • If   is a paranorm on   then so is the map  [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   and   are paranorms on   then so is   Moreover,   and   This makes the set of paranorms on   into a conditionally complete lattice.[8]
  • Each of the following real-valued maps are paranorms on  :
    •  
    •  
  • The real-valued maps   and   are not a paranorms on  [8]
  • If   is a Hamel basis on a vector space   then the real-valued map that sends   (where all but finitely many of the scalars   are 0) to   is a paranorm on   which satisfies   for all   and scalars  [8]
  • The function  is a paranorm on   that is not balanced but nevertheless equivalent to the usual norm on   Note that the function   is subadditive.[10]
  • Let   be a complex vector space and let   denote   considered as a vector space over   Any paranorm on   is also a paranorm on  [9]

F-seminorms edit

If   is a vector space over the real or complex numbers then an F-seminorm on   (the   stands for Fréchet) is a real-valued map   with the following four properties: [11]

  1. Non-negative:  
  2. Subadditive:   for all  
  3. Balanced:   for   all scalars   satisfying  
    • This condition guarantees that each set of the form   or   for some   is a balanced set.
  4. For every     as  
    • The sequence   can be replaced by any positive sequence converging to the zero.[12]

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

  1. Total/Positive definite:   implies  

An F-seminorm is called monotone if it satisfies:

  1. Monotone:   for all non-zero   and all real   and   such that  [12]

F-seminormed spaces edit

An F-seminormed space (resp. F-normed space)[12] is a pair   consisting of a vector space   and an F-seminorm (resp. F-norm)   on  

If   and   are F-seminormed spaces then a map   is called an isometric embedding[12] if  

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-seminorms edit

  • Every positive scalar multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm).
  • The sum of finitely many F-seminorms (resp. F-norms) is an F-seminorm (resp. F-norm).
  • If   and   are F-seminorms on   then so is their pointwise supremum   The same is true of the supremum of any non-empty finite family of F-seminorms on  [12]
  • The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).[9]
  • A non-negative real-valued function on   is a seminorm if and only if it is a 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   the map   on   defined by
     
    is an F-norm that is not a norm.
  • If   is a linear map and if   is an F-seminorm on   then   is an F-seminorm on  [12]
  • Let   be a complex vector space and let   denote   considered as a vector space over   Any F-seminorm on   is also an F-seminorm on  [9]

Properties of F-seminorms edit

Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.[7] Every F-seminorm on a vector space   is a value on   In particular,   and   for all  

Topology induced by a single F-seminorm edit

Theorem[11] — Let   be an F-seminorm on a vector space   Then the map   defined by   is a translation invariant pseudometric on   that defines a vector topology   on   If   is an F-norm then   is a metric. When   is endowed with this topology then   is a continuous map on  

The balanced sets   as   ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. Similarly, the balanced sets   as   ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets.

Topology induced by a family of F-seminorms edit

Suppose that   is a non-empty collection of F-seminorms on a vector space   and for any finite subset   and any   let

 

The set   forms a filter base on   that also forms a neighborhood basis at the origin for a vector topology on   denoted by  [12] Each   is a balanced and absorbing subset of  [12] These sets satisfy[12]

 
  •   is the coarsest vector topology on   making each   continuous.[12]
  •   is Hausdorff if and only if for every non-zero   there exists some   such that  [12]
  • If   is the set of all continuous F-seminorms on   then  [12]
  • If   is the set of all pointwise suprema of non-empty finite subsets of   of   then   is a directed family of F-seminorms and  [12]

Fréchet combination edit

Suppose that   is a family of non-negative subadditive functions on a vector space  

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

 

As an F-seminorm edit

Assume that   is an increasing sequence of seminorms on   and let   be the Fréchet combination of   Then   is an F-seminorm on   that induces the same locally convex topology as the family   of seminorms.[13]

Since   is increasing, a basis of open neighborhoods of the origin consists of all sets of the form   as   ranges over all positive integers and   ranges over all positive real numbers.

The translation invariant pseudometric on   induced by this F-seminorm   is

 

This metric was discovered by Fréchet in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.[14]

As a paranorm edit

If each   is a paranorm then so is   and moreover,   induces the same topology on   as the family   of paranorms.[8] This is also true of the following paranorms on  :

  •  [8]
  •  [8]

Generalization edit

The Fréchet combination can be generalized by use of a bounded remetrization function.

A bounded remetrization function[15] is a continuous non-negative non-decreasing map   that has a bounded range, is subadditive (meaning that   for all  ), and satisfies   if and only if  

Examples of bounded remetrization functions include       and  [15] If   is a pseudometric (respectively, metric) on   and   is a bounded remetrization function then   is a bounded pseudometric (respectively, bounded metric) on   that is uniformly equivalent to  [15]

Suppose that   is a family of non-negative F-seminorm on a vector space     is a bounded remetrization function, and   is a sequence of positive real numbers whose sum is finite. Then

 
defines a bounded F-seminorm that is uniformly equivalent to the  [16] It has the property that for any net   in     if and only if   for all  [16]   is an F-norm if and only if the   separate points on  [16]

Characterizations edit

Of (pseudo)metrics induced by (semi)norms edit

A pseudometric (resp. metric)   is induced by a seminorm (resp. norm) on a vector space   if and only if   is translation invariant and absolutely homogeneous, which means that for all scalars   and all   in which case the function defined by   is a seminorm (resp. norm) and the pseudometric (resp. metric) induced by   is equal to  

Of pseudometrizable TVS edit

If   is a topological vector space (TVS) (where note in particular that   is assumed to be a vector topology) then the following are equivalent:[11]

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

Of metrizable TVS edit

If   is a TVS then the following are equivalent:

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

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

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

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

Of locally convex pseudometrizable TVS edit

If   is TVS then the following are equivalent:[13]

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

Quotients edit

Let   be a vector subspace of a topological vector space  

  • If   is a pseudometrizable TVS then so is  [11]
  • If   is a complete pseudometrizable TVS and   is a closed vector subspace of   then   is complete.[11]
  • If   is metrizable TVS and   is a closed vector subspace of   then   is metrizable.[11]
  • If   is an F-seminorm on   then the map   defined by
     
    is an F-seminorm on   that induces the usual quotient topology on  [11] If in addition   is an F-norm on   and if   is a closed vector subspace of   then   is an F-norm on  [11]

Examples and sufficient conditions edit

  • Every seminormed space   is pseudometrizable with a canonical pseudometric given by   for all  [19].
  • If   is pseudometric TVS with a translation invariant pseudometric   then   defines a paranorm.[20] However, if   is a translation invariant pseudometric on the vector space   (without the addition condition that   is pseudometric TVS), then   need not be either an F-seminorm[21] nor a paranorm.
  • If a TVS has a bounded neighborhood of the origin 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   is either a DF-space or an LM-space. If   is a sequential space then it is either metrizable or else a Montel DF-space.

If   is Hausdorff locally convex TVS then   with the strong topology,   is metrizable if and only if there exists a countable set   of bounded subsets of   such that every bounded subset of   is contained in some element of  [22]

The strong dual space   of a metrizable locally convex space (such as a Fréchet space[23])   is a DF-space.[24] The strong dual of a DF-space is a Fréchet space.[25] The strong dual of a reflexive Fréchet space is a bornological space.[24] The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.[26] If   is a metrizable locally convex space then its strong dual   has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.[26]

Normability edit

A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin. Moreover, a TVS is normable if and only if it is Hausdorff and seminormable.[14] Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is not normable must be infinite dimensional.

If   is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then   is normable.[27]

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

  1.   is normable.
  2.   has a (von Neumann) bounded neighborhood of the origin.
  3. the strong dual space   of   is normable.[28]

and if this locally convex space   is also metrizable, then the following may be appended to this list:

  1. the strong dual space of   is metrizable.[28]
  2. the strong dual space of   is a Fréchet–Urysohn locally convex space.[23]

In particular, if a metrizable locally convex space   (such as a Fréchet space) is not normable then its strong dual space   is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space   is also neither metrizable nor normable.

Another consequence of this is that if   is a reflexive locally convex TVS whose strong dual   is metrizable then   is necessarily a reflexive Fréchet space,   is a DF-space, both   and   are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover,   is normable if and only if   is normable if and only if   is Fréchet–Urysohn if and only if   is metrizable. In particular, such a space   is either a Banach space or else it is not even a Fréchet–Urysohn space.

Metrically bounded sets and bounded sets edit

Suppose that   is a pseudometric space and   The set   is metrically bounded or  -bounded if there exists a real number   such that   for all  ; the smallest such   is then called the diameter or  -diameter of  [14] If   is bounded in a pseudometrizable TVS   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 edit

Theorem[29] — All infinite-dimensional separable complete metrizable TVS are homeomorphic.

  • Every metrizable locally convex TVS is a quasibarrelled space,[30] bornological space, and a Mackey space.
  • Every complete pseudometrizable TVS is a barrelled space and a Baire space (and hence non-meager).[31] However, there exist metrizable Baire spaces that are not complete.[31]
  • If   is a metrizable locally convex space, then the strong dual of   is bornological if and only if it is barreled, if and only if it is infrabarreled.[26]
  • If   is a complete pseudometrizable TVS and   is a closed vector subspace of   then   is complete.[11]
  • The strong dual of a locally convex metrizable TVS is a webbed space.[32]
  • If   and   are complete metrizable TVSs (i.e. F-spaces) and if   is coarser than   then  ;[33] this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete.[34] Said differently, if   and   are both F-spaces but with different topologies, then neither one of   and   contains the other as a subset. One particular consequence of this is, for example, that if   is a Banach space and   is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of   (i.e. if   or if   for some constant  ), then the only way that   can be a Banach space (i.e. also be complete) is if these two norms   and   are equivalent; if they are not equivalent, then   can not be a Banach space. As another consequence, if   is a Banach space and   is a Fréchet space, then the map   is continuous if and only if the Fréchet space   is the TVS   (here, the Banach space   is being considered as a TVS, which means that its norm is "forgetten" but its topology is remembered).
  • A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.[23]
  • Any product of complete metrizable TVSs is a Baire space.[31]
  • A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension  [35]
  • 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 pseudometrizable TVS is a barrelled space and a Baire space (and thus non-meager).[31]
  • The dimension of a complete metrizable TVS is either finite or uncountable.[35]

Completeness edit

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   is a metrizable TVS and   is a metric that defines  's topology, then its possible that   is complete as a TVS (i.e. relative to its uniformity) but the metric   is not a complete metric (such metrics exist even for  ). Thus, if   is a TVS whose topology is induced by a pseudometric   then the notion of completeness of   (as a TVS) and the notion of completeness of the pseudometric space   are not always equivalent. The next theorem gives a condition for when they are equivalent:

Theorem — If   is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric   then   is a complete pseudometric on   if and only if   is complete as a TVS.[36]

Theorem[37][38] (Klee) — Let   be any[note 2] metric on a vector space   such that the topology   induced by   on   makes   into a topological vector space. If   is a complete metric space then   is a complete-TVS.

Theorem — If   is a TVS whose topology is induced by a paranorm   then   is complete if and only if for every sequence   in   if   then   converges in  [39]

If   is a closed vector subspace of a complete pseudometrizable TVS   then the quotient space   is complete.[40] If   is a complete vector subspace of a metrizable TVS   and if the quotient space   is complete then so is  [40] If   is not complete then   but not complete, vector subspace of  

A Baire separable topological group is metrizable if and only if it is cosmic.[23]

Subsets and subsequences edit

  • Let   be a separable locally convex metrizable topological vector space and let   be its completion. If   is a bounded subset of   then there exists a bounded subset   of   such that  [41]
  • Every totally bounded subset of a locally convex metrizable TVS   is contained in the closed convex balanced hull of some sequence in   that converges to  
  • In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.[42]
  • If   is a translation invariant metric on a vector space   then   for all   and every positive integer  [43]
  • If   is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence   of positive real numbers diverging to   such that  [43]
  • A subset of a complete metric space is closed if and only if it is complete. If a space   is not complete, then   is a closed subset of   that is not complete.
  • If   is a metrizable locally convex TVS then for every bounded subset   of   there exists a bounded disk   in   such that   and both   and the auxiliary normed space   induce the same subspace topology on  [44]

Banach-Saks theorem[45] — If   is a sequence in a locally convex metrizable TVS   that converges weakly to some   then there exists a sequence   in   such that   in   and each   is a convex combination of finitely many  

Mackey's countability condition[14] — Suppose that   is a locally convex metrizable TVS and that   is a countable sequence of bounded subsets of   Then there exists a bounded subset   of   and a sequence   of positive real numbers such that   for all  

Generalized series

As described in this article's section on generalized series, for any  -indexed family family   of vectors from a TVS   it is possible to define their sum   as the limit of the net of finite partial sums   where the domain   is directed by   If   and   for instance, then the generalized series   converges if and only if   converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series   converges in a metrizable TVS, then the set   is necessarily countable (that is, either finite or countably infinite);[proof 1] in other words, all but at most countably many   will be zero and so this generalized series   is actually a sum of at most countably many non-zero terms.

Linear maps edit

If   is a pseudometrizable TVS and   maps bounded subsets of   to bounded subsets of   then   is continuous.[14] Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.[46] Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.[46]

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

  1.   is continuous;
  2.   is a (locally) bounded map (that is,   maps (von Neumann) bounded subsets of   to bounded subsets of  );[12]
  3.   is sequentially continuous;[12]
  4. the image under   of every null sequence in   is a bounded set[12] where by definition, a null sequence is a sequence that converges to the origin.
  5.   maps null sequences to null sequences;

Open and almost open maps

Theorem: If   is a complete pseudometrizable TVS,   is a Hausdorff TVS, and