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{{Short description|A topological vector space whose topology can be defined by a metric}}
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 space|pseudometric]]). An '''LM-space''' is an [[inductive limit]] of a sequence of [[locally convex]] metrizable TVS.

==Pseudometrics and metrics==

A '''[[Pseudometric space|pseudometric]]''' on a set <math>X</math> is a map <math>d : X \times X \rarr \R</math> satisfying the following properties:
<ol>
<li><math>d(x, x) = 0 \text{ for all } x \in X</math>;</li>
<li>'''Symmetry''': <math>d(x, y) = d(y, x) \text{ for all } x, y \in X</math>;</li>
<li>'''[[Subadditive|Subadditivity]]''': <math>d(x, z) \leq d(x, y) +  d(y, z) \text{ for all } x, y, z \in X.</math></li>
</ol>

A pseudometric is called a '''[[Metric (mathematics)|metric]]''' if it satisfies: 
<ol start=4>
<li>'''[[Identity of indiscernibles]]''': for all <math>x, y \in X,</math> if <math>d(x, y) = 0</math> then <math>x = y.</math></li>
</ol>

'''Ultrapseudometric'''

A pseudometric <math>d</math> on <math>X</math> is called a '''[[Ultrametric|ultrapseudometric]]''' or a '''strong pseudometric''' if it satisfies: 
<ol start=5>
<li>'''Strong'''/'''Ultrametric triangle inequality''': <math>d(x, z) \leq \max \{ d(x, y),  d(y, z) \} \text{ for all } x, y, z \in X.</math></li>
</ol>

'''Pseudometric space'''

A '''[[pseudometric space]]''' is a pair <math>(X, d)</math> consisting of a set <math>X</math> and a pseudometric <math>d</math> on <math>X</math> such that <math>X</math>'s topology is identical to the topology on <math>X</math> induced by <math>d.</math> We call a pseudometric space <math>(X, d)</math> a '''[[metric space]]''' (resp. '''ultrapseudometric space''') when <math>d</math> is a metric (resp. ultrapseudometric). 

===Topology induced by a pseudometric===

If <math>d</math> is a pseudometric on a set <math>X</math> then collection of '''open balls''':
<math display="block">B_r(z) := \{ x \in X : d(x, z) < r \}</math> as <math>z</math> ranges over <math>X</math> and <math>r > 0</math> ranges over the positive real numbers,
forms a basis for a topology on <math>X</math> that is called the <math>d</math>'''-topology''' or the '''pseudometric topology''' on <math>X</math> induced by <math>d.</math> 

:{{em|Convention}}: If <math>(X, d)</math> is a pseudometric space and <math>X</math> is treated as a [[topological space]], then unless indicated otherwise, it should be assumed that <math>X</math> is endowed with the topology induced by <math>d.</math> 

'''Pseudometrizable space'''

A topological space <math>(X, \tau)</math> is called '''pseudometrizable''' (resp. '''metrizable''', '''ultrapseudometrizable''') if there exists a pseudometric (resp. metric, ultrapseudometric) <math>d</math> on <math>X</math> such that <math>\tau</math> is equal to the topology induced by <math>d.</math>{{sfn|Narici|Beckenstein|2011|pp=1-18}}

==Pseudometrics and values on topological groups==

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 <math>\tau</math> on a real or complex vector space <math>X</math> 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 <math>X</math> into a [[topological vector space]]). 

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

===Translation invariant pseudometrics===

If <math>X</math> is an additive group then we say that a pseudometric <math>d</math> on <math>X</math> is '''translation invariant''' or just '''invariant''' if it satisfies any of the following equivalent conditions:
<ol>
<li>'''[[Translation invariance]]''': <math>d(x + z, y + z) = d(x, y) \text{ for all } x, y, z \in X</math>;</li>
<li><math>d(x, y) = d(x - y, 0) \text{ for all } x, y \in X.</math></li>
</ol>

===Value/G-seminorm===

If <math>X</math> is a [[topological group]] the a '''value''' or '''G-seminorm''' on <math>X</math> (the ''G'' stands for Group) is a real-valued map <math>p : X \rarr \R</math> with the following properties:{{sfn|Narici|Beckenstein|2011|pp=37-40}}
<ol>
<li>'''Non-negative''': <math>p \geq 0.</math></li>
<li>'''Subadditive''': <math>p(x + y) \leq p(x) + p(y) \text{ for all } x, y \in X</math>;</li>
<li><math>p(0) = 0..</math></li>
<li>'''Symmetric''': <math>p(-x) = p(x) \text{ for all } x \in X.</math></li>
</ol>

where we call a G-seminorm a '''G-norm''' if it satisfies the additional condition:
<ol start=5>
<li>'''Total'''/'''Positive definite''': If <math>p(x) = 0</math> then <math>x = 0.</math></li>
</ol>

====Properties of values====

If <math>p</math> is a value on a vector space <math>X</math> then:
<ul>
<li><math>|p(x) - p(y)| \leq p(x - y) \text{ for all } x, y \in X.</math>{{sfn|Swartz|1992|p=15}}</li>
<li><math>p(n x) \leq n p(x)</math> and <math>\frac{1}{n} p(x) \leq p(x / n)</math> for all <math>x \in X</math> and positive integers <math>n.</math>{{sfn|Wilansky|2013|p=17}}</li>
<li>The set <math>\{ x \in X : p(x) = 0 \}</math> is an additive subgroup of <math>X.</math>{{sfn|Swartz|1992|p=15}}</li>
</ul>

===Equivalence on topological groups===

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=37-40}}|math_statement=
Suppose that <math>X</math> is an additive commutative group. 
If <math>d</math> is a translation invariant pseudometric on <math>X</math> then the map <math>p(x) := d(x, 0)</math> is a value on <math>X</math> called '''the value associated with <math>d</math>''', and moreover, <math>d</math> generates a group topology on <math>X</math> (i.e. the <math>d</math>-topology on <math>X</math> makes <math>X</math> into a topological group).
Conversely, if <math>p</math> is a value on <math>X</math> then the map <math>d(x, y) := p(x - y)</math> is a translation-invariant pseudometric on <math>X</math> and the value associated with <math>d</math> is just <math>p.</math>
}}

===Pseudometrizable topological groups===

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=37-40}}|math_statement=
If <math>(X, \tau)</math> is an additive commutative [[topological group]] then the following are equivalent:
<ol>
<li><math>\tau</math> is induced by a pseudometric; (i.e. <math>(X, \tau)</math> is pseudometrizable);</li>
<li><math>\tau</math> is induced by a translation-invariant pseudometric;</li>
<li>the identity element in <math>(X, \tau)</math> has a countable neighborhood basis.</li>
</ol>
If <math>(X, \tau)</math> is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric." 
A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable. 
}}

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

Let <math>X</math> be a non-trivial (i.e. <math>X \neq \{ 0 \}</math>) real or complex vector space and let <math>d</math> be the translation-invariant '''trivial metric''' on <math>X</math> defined by <math>d(x, x) = 0</math> and <math>d(x, y) = 1 \text{ for all } x, y \in X</math> such that <math>x \neq y.</math> 
The topology <math>\tau</math> that <math>d</math> induces on <math>X</math> is the [[discrete topology]], which makes <math>(X, \tau)</math> into a commutative topological group under addition but does {{em|not}} form a vector topology on <math>X</math> because <math>(X, \tau)</math> is [[Connected space|disconnected]] but every vector topology is connected. 
What fails is that scalar multiplication isn't continuous on <math>(X, \tau).</math> 

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

==Additive sequences==

A collection <math>\mathcal{N}</math> of subsets of a vector space is called '''additive'''{{sfn|Wilansky|2013|pp=40-47}} if for every <math>N \in \mathcal{N},</math> there exists some <math>U \in \mathcal{N}</math> such that <math>U + U \subseteq N.</math>

{{Math theorem|name=Continuity of addition at 0|math_statement=
If <math>(X, +)</math> is a [[Group (mathematics)|group]] (as all vector spaces are), <math>\tau</math> is a topology on <math>X,</math> and <math>X \times X</math> is endowed with the [[product topology]], then the addition map <math>X \times X \to X</math> (i.e. the map <math>(x, y) \mapsto x + y</math>) is continuous at the origin of <math>X \times X</math> if and only if the set of [[Neighborhood (topology)|neighborhood]]s of the origin in <math>(X, \tau)</math> is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."{{sfn|Wilansky|2013|pp=40-47}}
}}

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 group]]s. 

{{Math theorem|name=Theorem|math_statement=
Let <math>U_{\bull} = \left(U_i\right)_{i=0}^{\infty}</math> be a collection of subsets of a vector space such that <math>0 \in U_i</math> and <math>U_{i+1} + U_{i+1} \subseteq U_i</math> for all <math>i \geq 0.</math> 
For all <math>u \in U_0,</math> let 
<math display="block">\mathbb{S}(u) := \left\{ n_{\bull} = \left(n_1, \ldots, n_k\right) ~:~ k \geq 1, n_i \geq 0 \text{ for all } i, \text{ and } u \in U_{n_1} + \cdots + U_{n_k}\right\}.</math>  

Define <math>f : X \to [0, 1]</math> by <math>f(x) = 1</math> if <math>x \not\in U_0</math> and otherwise let 
<math display="block">f(x) := \inf_{} \left\{ 2^{- n_1} + \cdots 2^{- n_k} ~:~ n_{\bull} = \left(n_1, \ldots, n_k\right) \in \mathbb{S}(x)\right\}.</math> 

Then <math>f</math> is [[Subadditivity|subadditive]] (meaning <math>f(x + y) \leq f(x) + f(y) \text{ for all } x, y \in X</math>) and <math>f = 0</math> on <math>\bigcap_{i \geq 0} U_i,</math> so in particular <math>f(0) = 0.</math> 
If all <math>U_i</math> are [[symmetric set]]s then <math>f(-x) = f(x)</math> and if all <math>U_i</math> are balanced then <math>f(s x) \leq f(x)</math> for all scalars <math>s</math> such that <math>|s| \leq 1</math> and all <math>x \in X.</math> 
If <math>X</math> is a topological vector space and if all <math>U_i</math> are neighborhoods of the origin then <math>f</math> is continuous, where if in addition <math>X</math> is Hausdorff and <math>U_{\bull}</math> forms a basis of balanced neighborhoods of the origin in <math>X</math> then <math>d(x, y) := f(x - y)</math> is a metric defining the vector topology on <math>X.</math> 
}}

{{collapse top|title=Proof|left=true}}

Assume that <math>n_{\bull} = \left(n_1, \ldots, n_k\right)</math> always denotes a finite sequence of non-negative integers and use the notation: 
<math display="block">\sum 2^{- n_{\bull}} := 2^{- n_1} + \cdots + 2^{- n_k} \quad \text{ and } \quad \sum U_{n_{\bull}} := U_{n_1} + \cdots + U_{n_k}.</math> 

For any integers <math>n \geq 0</math> and <math>d > 2,</math> 
<math display="block">U_n \supseteq U_{n+1} + U_{n+1} \supseteq U_{n+1} + U_{n+2} + U_{n+2} \supseteq U_{n+1} + U_{n+2} + \cdots + U_{n+d} + U_{n+d+1} + U_{n+d+1}.</math> 

From this it follows that if <math>n_{\bull} = \left(n_1, \ldots, n_k\right)</math> consists of distinct positive integers then <math>\sum U_{n_{\bull}} \subseteq U_{-1 + \min \left(n_{\bull}\right)}.</math> 

It will now be shown by induction on <math>k</math> that if <math>n_{\bull} = \left(n_1, \ldots, n_k\right)</math> consists of non-negative integers such that <math>\sum 2^{- n_{\bull}} \leq 2^{- M}</math> for some integer <math>M \geq 0</math> then <math>\sum U_{n_{\bull}} \subseteq U_M.</math> 
This is clearly true for <math>k = 1</math> and <math>k = 2</math> so assume that <math>k > 2,</math> which implies that all <math>n_i</math> are positive. 
If all <math>n_i</math> are distinct then this step is done, and otherwise pick distinct indices <math>i < j</math> such that <math>n_i = n_j</math> and construct <math>m_{\bull} = \left(m_1, \ldots, m_{k-1}\right)</math> from <math>n_{\bull}</math> by replacing each <math>n_i</math> with <math>n_i - 1</math> and deleting the <math>j^{\text{th}}</math> element of <math>n_{\bull}</math> (all other elements of <math>n_{\bull}</math> are transferred to <math>m_{\bull}</math> unchanged). 
Observe that <math>\sum 2^{- n_{\bull}} = \sum 2^{- m_{\bull}}</math> and <math>\sum U_{n_{\bull}} \subseteq \sum U_{m_{\bull}}</math> (because <math>U_{n_i} + U_{n_j} \subseteq U_{n_i - 1}</math>) so by appealing to the inductive hypothesis we conclude that <math>\sum U_{n_{\bull}} \subseteq \sum U_{m_{\bull}} \subseteq U_M,</math> as desired. 

It is clear that <math>f(0) = 0</math> and that <math>0 \leq f \leq 1</math> so to prove that <math>f</math> is subadditive, it suffices to prove that <math>f(x + y) \leq f(x) + f(y)</math> when <math>x, y \in X</math> are such that <math>f(x) + f(y) < 1,</math> which implies that <math>x, y \in U_0.</math> 
This is an exercise. 
If all <math>U_i</math> are symmetric then <math>x \in \sum U_{n_{\bull}}</math> if and only if <math>- x \in \sum U_{n_{\bull}}</math> from which it follows that <math>f(-x) \leq f(x)</math> and <math>f(-x) \geq f(x).</math> 
If all <math>U_i</math> are balanced then the inequality <math>f(s x) \leq f(x)</math> for all unit scalars <math>s</math> such that <math>|s| \leq 1</math> is proved similarly. 
Because <math>f</math> is a nonnegative subadditive function satisfying <math>f(0) = 0,</math> as described in the article on [[sublinear functional]]s, <math>f</math> is uniformly continuous on <math>X</math> if and only if <math>f</math> is continuous at the origin. 
If all <math>U_i</math> are neighborhoods of the origin then for any real <math>r > 0,</math> pick an integer <math>M > 1</math> such that <math>2^{-M} < r</math> so that <math>x \in U_M</math> implies <math>f(x) \leq 2^{-M} < r.</math> 
If the set of all <math>U_i</math> form basis of balanced neighborhoods of the origin then it may be shown that for any <math>n > 1,</math> there exists some <math>0 < r \leq 2^{-n}</math> such that <math>f(x) < r</math> implies <math>x \in U_n.</math> <math>\blacksquare</math>
{{collapse bottom}}

==Paranorms==

If <math>X</math> is a vector space over the real or complex numbers then a '''paranorm''' on <math>X</math> is a G-seminorm (defined above) <math>p : X \rarr \R</math> on <math>X</math> that satisfies any of the following additional conditions, each of which begins with "for all sequences <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> in <math>X</math> and all convergent sequences of scalars <math>s_{\bull} = \left(s_i\right)_{i=1}^{\infty}</math>":{{sfn|Wilansky|2013|p=15}}
<ol>
<li>'''Continuity of multiplication''': if <math>s</math> is a scalar and <math>x \in X</math> are such that <math>p\left(x_i - x\right) \to 0</math> and <math>s_{\bull} \to s,</math> then <math>p\left(s_i x_i - s x\right) \to 0.</math></li>
<li>Both of the conditions:
* if <math>s_{\bull} \to 0</math> and if <math>x \in X</math> is such that <math>p\left(x_i - x\right) \to 0</math> then <math>p\left(s_i x_i\right) \to 0</math>;
* if <math>p\left(x_{\bull}\right) \to 0</math> then <math>p\left(s x_i\right) \to 0</math> for every scalar <math>s.</math></li>
<li>Both of the conditions:
* if <math>p\left(x_{\bull}\right) \to 0</math> and <math>s_{\bull} \to s</math> for some scalar <math>s</math> then <math>p\left(s_i x_i\right) \to 0</math>;
* if <math>s_{\bull} \to 0</math> then <math>p\left(s_i x\right) \to 0 \text{ for all } x \in X.</math></li>
<li>'''Separate continuity''':{{sfn|Schechter|1996|pp=689-691}}
* if <math>s_{\bull} \to s</math> for some scalar <math>s</math> then <math>p\left(s x_i - s x\right) \to 0</math> for every <math>x \in X</math>;
* if <math>s</math> is a scalar, <math>x \in X,</math> and <math>p\left(x_i - x\right) \to 0</math> then <math>p\left(s x_i - s x\right) \to 0</math> .</li>
</ol>

A paranorm is called '''total''' if in addition it satisfies:
<ul>
<li>'''Total'''/'''Positive definite''': <math>p(x) = 0</math> implies <math>x = 0.</math></li>
</ul>

===Properties of paranorms===

If <math>p</math> is a paranorm on a vector space <math>X</math> then the map <math>d : X \times X \rarr \R</math> defined by <math>d(x, y) := p(x - y)</math> is a translation-invariant pseudometric on <math>X</math> that defines a {{em|vector topology}} on <math>X.</math>{{sfn|Wilansky|2013|pp=15-18}}

If <math>p</math> is a paranorm on a vector space <math>X</math> then:
<ul>
<li>the set <math>\{ x \in X : p(x) = 0 \}</math> is a vector subspace of <math>X.</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
<li><math>p(x + n) = p(x) \text{ for all } x, n \in X</math> with <math>p(n) = 0.</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>If a paranorm <math>p</math> satisfies <math>p(s x) \leq |s| p(x) \text{ for all } x \in X</math> and scalars <math>s,</math> then <math>p</math> is absolutely homogeneity (i.e. equality holds){{sfn|Wilansky|2013|pp=15-18}} and thus <math>p</math> is a [[seminorm]].</li>
</ul>

===Examples of paranorms===

<ul>
<li>If <math>d</math> is a translation-invariant pseudometric on a vector space <math>X</math> that induces a vector topology <math>\tau</math> on <math>X</math> (i.e. <math>(X, \tau)</math> is a TVS) then the map <math>p(x) := d(x - y, 0)</math> defines a continuous paranorm on <math>(X, \tau)</math>; moreover, the topology that this paranorm <math>p</math> defines in <math>X</math> is <math>\tau.</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>If <math>p</math> is a paranorm on <math>X</math> then so is the map <math>q(x) := p(x) / [1 + p(x)].</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).</li>
<li>Every [[seminorm]] is a paranorm.{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).{{sfn|Schechter|1996|p=692}}</li>
<li>The sum of two paranorms is a paranorm.{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>If <math>p</math> and <math>q</math> are paranorms on <math>X</math> then so is <math>(p \wedge q)(x) := \inf_{} \{ p(y) + q(z) : x = y + z \text{ with } y, z \in X \}.</math> Moreover, <math>(p \wedge q) \leq p</math> and <math>(p \wedge q) \leq q.</math> This makes the set of paranorms on <math>X</math> into a [[Lattice_(order)#Conditional_completeness|conditionally complete lattice]].{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>Each of the following real-valued maps are paranorms on <math>X := \R^2</math>:
* <math>(x, y) \mapsto |x|</math> 
* <math>(x, y) \mapsto |x| + |y|</math></li>
<li>The real-valued maps <math>(x, y) \mapsto \sqrt{\left|x^2 - y^2\right|}</math> and <math>(x, y) \mapsto \left|x^2 - y^2\right|^{3/2}</math> are {{em|not}} paranorms on <math>X := \R^2.</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a [[Hamel basis]] on a vector space <math>X</math> then the real-valued map that sends <math>x = \sum_{i \in I} s_i x_i \in X</math> (where all but finitely many of the scalars <math>s_i</math> are 0) to <math>\sum_{i \in I} \sqrt{\left|s_i\right|}</math> is a paranorm on <math>X,</math> which satisfies <math>p(sx) = \sqrt{|s|} p(x)</math> for all <math>x \in X</math> and scalars <math>s.</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
<li>The function <math>p(x) := |\sin (\pi x)| + \min \{ 2, |x| \}</math> is a paranorm on <math>\R</math> that is {{em|not}} balanced but nevertheless equivalent to the usual norm on <math>R.</math> Note that the function <math>x \mapsto |\sin (\pi x)|</math> is subadditive.{{sfn|Schechter|1996|p=691}}</li>
<li>Let <math>X_{\Complex}</math> be a complex vector space and let <math>X_{\R}</math> denote <math>X_{\Complex}</math> considered as a vector space over <math>\R.</math> Any paranorm on <math>X_{\Complex}</math> is also a paranorm on <math>X_{\R}.</math>{{sfn|Schechter|1996|p=692}}</li>
<li>
</ul>

==''F''-seminorms==

If <math>X</math> is a vector space over the real or complex numbers then an '''''F''-seminorm''' on <math>X</math> (the <math>F</math> stands for [[Maurice René Fréchet|Fréchet]]) is a real-valued map <math>p : X \to \Reals</math> with the following four properties: {{sfn|Narici|Beckenstein|2011|pp=91-95}} 
<ol>
<li>'''Non-negative''': <math>p \geq 0.</math></li>
<li>'''Subadditive''': <math>p(x + y) \leq p(x) + p(y)</math> for all <math>x, y \in X</math></li>
<li>'''[[Balanced function|Balanced]]''': <math>p(a x) \leq p(x)</math> for <math>x \in X</math> all scalars <math>a</math> satisfying <math>|a| \leq 1;</math>
* This condition guarantees that each set of the form <math>\{z \in X : p(z) \leq r\}</math> or <math>\{z \in X : p(z) < r\}</math> for some <math>r \geq 0</math> is a [[balanced set]].</li>
<li>For every <math>x \in X,</math> <math>p\left(\tfrac{1}{n} x\right) \to 0</math> as <math>n \to \infty</math>
* The sequence <math>\left(\tfrac{1}{n}\right)_{n=1}^\infty</math> can be replaced by any positive sequence converging to the zero.{{sfn|Jarchow|1981|pp=38-42}}</li>
</ol>

An ''F''-seminorm is called an '''''F''-norm''' if in addition it satisfies:
<ol start=5>
<li>'''Total'''/'''Positive definite''': <math>p(x) = 0</math> implies <math>x = 0.</math></li>
</ol>

An ''F''-seminorm is called '''monotone''' if it satisfies:
<ol start=6>
<li>'''Monotone''': <math>p(r x) < p(s x)</math> for all non-zero <math>x \in X</math> and all real <math>s</math> and <math>t</math> such that <math>s < t.</math>{{sfn|Jarchow|1981|pp=38-42}}</li>
</ol>

===''F''-seminormed spaces===

An '''''F''-seminormed space''' (resp. '''''F''-normed space'''){{sfn|Jarchow|1981|pp=38-42}} is a pair <math>(X, p)</math> consisting of a vector space <math>X</math> and an ''F''-seminorm (resp. ''F''-norm) <math>p</math> on <math>X.</math> 

If <math>(X, p)</math> and <math>(Z, q)</math> are ''F''-seminormed spaces then a map <math>f : X \to Z</math> is called an '''isometric embedding'''{{sfn|Jarchow|1981|pp=38-42}} if <math>q(f(x) - f(y)) = p(x, y) \text{ for all } x, y \in X.</math> 

Every isometric embedding of one ''F''-seminormed space into another is a [[topological embedding]], but the converse is not true in general.{{sfn|Jarchow|1981|pp=38-42}} 

===Examples of ''F''-seminorms===

<ul>
<li>Every positive scalar multiple of an ''F''-seminorm (resp. ''F''-norm, seminorm) is again an ''F''-seminorm (resp. ''F''-norm, seminorm).</li>
<li>The sum of finitely many ''F''-seminorms (resp. ''F''-norms) is an ''F''-seminorm (resp. ''F''-norm).</li>
<li>If <math>p</math> and <math>q</math> are ''F''-seminorms on <math>X</math> then so is their pointwise supremum <math>x \mapsto \sup \{p(x), q(x)\}.</math> The same is true of the supremum of any non-empty finite family of ''F''-seminorms on <math>X.</math>{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>The restriction of an ''F''-seminorm (resp. ''F''-norm) to a vector subspace is an ''F''-seminorm (resp. ''F''-norm).{{sfn|Schechter|1996|p=692}}</li>
<li>A non-negative real-valued function on <math>X</math> is a seminorm if and only if it is a [[Convex function|convex]] ''F''-seminorm, or equivalently, if and only if it is a convex balanced ''G''-seminorm.{{sfn|Schechter|1996|p=691}}  In particular, every [[seminorm]] is an ''F''-seminorm.</li>
<li>For any <math>0 < p < 1,</math> the map <math>f</math> on <math>\Reals^n</math> defined by 
<math display="block">[f\left(x_1, \ldots, x_n\right)]^p = \left|x_1\right|^p + \cdots \left|x_n\right|^p</math> 
is an ''F''-norm that is not a norm.</li>
<li>If <math>L : X \to Y</math> is a linear map and if <math>q</math> is an ''F''-seminorm on <math>Y,</math> then <math>q \circ L</math> is an ''F''-seminorm on <math>X.</math>{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>Let <math>X_\Complex</math> be a complex vector space and let <math>X_\Reals</math> denote <math>X_\Complex</math> considered as a vector space over <math>\Reals.</math> Any ''F''-seminorm on <math>X_\Complex</math> is also an ''F''-seminorm on <math>X_\Reals.</math>{{sfn|Schechter|1996|p=692}}</li>
</ul>

===Properties of ''F''-seminorms===

Every ''F''-seminorm is a paranorm and every paranorm is equivalent to some ''F''-seminorm.{{sfn|Schechter|1996|pp=689-691}}  
Every ''F''-seminorm on a vector space <math>X</math> is a value on <math>X.</math> In particular, <math>p(x) = 0,</math> and <math>p(x) = p(-x)</math> for all <math>x \in X.</math>  

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

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=91-95}}|math_statement=
Let <math>p</math> be an ''F''-seminorm on a vector space <math>X.</math> 
Then the map <math>d : X \times X \to \Reals</math> defined by 
<math>d(x, y) := p(x - y)</math> 
is a translation invariant pseudometric on <math>X</math> that defines a vector topology <math>\tau</math> on <math>X.</math> 
If <math>p</math> is an ''F''-norm then <math>d</math> is a metric. 
When <math>X</math> is endowed with this topology then <math>p</math> is a continuous map on <math>X.</math> 

The balanced sets <math>\{x \in X ~:~ p(x) \leq r\},</math> as <math>r</math> ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. 
Similarly, the balanced sets <math>\{x \in X ~:~ p(x) < r\},</math> as <math>r</math> ranges over the positive reals, form a neighborhood basis at the origin for this topology consisting of open sets. 
}}

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

Suppose that <math>\mathcal{L}</math> is a non-empty collection of ''F''-seminorms on a vector space <math>X</math> and for any finite subset <math>\mathcal{F} \subseteq \mathcal{L}</math> and any <math>r > 0,</math> let 
<math display="block">U_{\mathcal{F}, r} := \bigcap_{p \in \mathcal{F}} \{x \in X : p(x) < r\}.</math> 

The set <math>\left\{U_{\mathcal{F}, r} ~:~ r > 0, \mathcal{F} \subseteq \mathcal{L}, \mathcal{F} \text{ finite }\right\}</math> forms a filter base on <math>X</math> that also forms a neighborhood basis at the origin for a vector topology on <math>X</math> denoted by <math>\tau_{\mathcal{L}}.</math>{{sfn|Jarchow|1981|pp=38-42}}  Each <math>U_{\mathcal{F}, r}</math> is a [[Balanced set|balanced]] and [[Absorbing set|absorbing]] subset of <math>X.</math>{{sfn|Jarchow|1981|pp=38-42}}  These sets satisfy{{sfn|Jarchow|1981|pp=38-42}}
<math display="block">U_{\mathcal{F}, r/2} + U_{\mathcal{F}, r/2} \subseteq U_{\mathcal{F}, r}.</math>

<ul>
<li><math>\tau_{\mathcal{L}}</math> is the coarsest vector topology on <math>X</math> making each <math>p \in \mathcal{L}</math> continuous.{{sfn|Jarchow|1981|pp=38-42}}</li>
<li><math>\tau_{\mathcal{L}}</math> is Hausdorff if and only if for every non-zero <math>x \in X,</math> there exists some <math>p \in \mathcal{L}</math> such that <math>p(x) > 0.</math>{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>If <math>\mathcal{F}</math> is the set of all continuous ''F''-seminorms on <math>\left(X, \tau_{\mathcal{L}}\right)</math> then <math>\tau_{\mathcal{L}} = \tau_{\mathcal{F}}.</math>{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>If <math>\mathcal{F}</math> is the set of all pointwise suprema of non-empty finite subsets of <math>\mathcal{F}</math> of <math>\mathcal{L}</math> then <math>\mathcal{F}</math> is a directed family of ''F''-seminorms and <math>\tau_{\mathcal{L}} = \tau_{\mathcal{F}}.</math>{{sfn|Jarchow|1981|pp=38-42}}</li>
</ul>

{{anchor|Fréchet combination}}

==Fréchet combination==

Suppose that <math>p_{\bull} = \left(p_i\right)_{i=1}^{\infty}</math> is a family of non-negative subadditive functions on a vector space <math>X.</math> 

The '''Fréchet combination'''{{sfn|Wilansky|2013|pp=15-18}} of <math>p_{\bull}</math> is defined to be the real-valued map 
<math display="block">p(x) := \sum_{i=1}^{\infty} \frac{p_i(x)}{2^{i} \left[ 1 + p_i(x)\right]}.</math>  

===As an ''F''-seminorm===

Assume that <math>p_{\bull} = \left(p_i\right)_{i=1}^{\infty}</math> is an increasing sequence of seminorms on <math>X</math> and let <math>p</math> be the Fréchet combination of <math>p_{\bull}.</math> 
Then <math>p</math> is an ''F''-seminorm on <math>X</math> that induces the same locally convex topology as the family <math>p_{\bull}</math> of seminorms.{{sfn|Narici|Beckenstein|2011|p=123}} 

Since <math>p_{\bull} = \left(p_i\right)_{i=1}^{\infty}</math> is increasing, a basis of open neighborhoods of the origin consists of all sets of the form <math>\left\{ x \in X ~:~ p_i(x) < r\right\}</math> as <math>i</math> ranges over all positive integers and <math>r > 0</math> ranges over all positive real numbers. 

The [[translation invariant]] [[Pseudometric space|pseudometric]] on <math>X</math> induced by this ''F''-seminorm <math>p</math> is 
<math display="block">d(x, y) = \sum^{\infty}_{i=1} \frac{1}{2^i} \frac{p_i( x - y )}{1 + p_i( x - y )}.</math>

This metric was discovered by [[Fréchet]] in his 1906 thesis for the spaces of real and complex sequences with pointwise operations.{{sfn|Narici|Beckenstein|2011|pp=156-175}}  

===As a paranorm===

If each <math>p_i</math> is a paranorm then so is <math>p</math> and moreover, <math>p</math> induces the same topology on <math>X</math> as the family <math>p_{\bull}</math> of paranorms.{{sfn|Wilansky|2013|pp=15-18}} 
This is also true of the following paranorms on <math>X</math>:
<ul>
<li><math>q(x) := \inf_{} \left\{ \sum_{i=1}^n p_i(x) + \frac{1}{n} ~:~ n > 0 \text{ is an integer }\right\}.</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
<li><math>r(x) := \sum_{n=1}^{\infty} \min \left\{ \frac{1}{2^n}, p_n(x)\right\}.</math>{{sfn|Wilansky|2013|pp=15-18}}</li>
</ul>

===Generalization===

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

A {{em|{{visible anchor|bounded remetrization function}}}}{{sfn|Schechter|1996|p=487}} is a continuous non-negative non-decreasing map <math>R : [0, \infty) \to [0, \infty)</math> that has a bounded range, is [[Subadditivity|subadditive]] (meaning that  <math>R(s + t) \leq R(s) + R(t)</math> for all <math>s, t \geq 0</math>), and satisfies <math>R(s) = 0</math> if and only if <math>s = 0.</math> 

Examples of bounded remetrization functions include <math>\arctan t,</math> <math>\tanh t,</math> <math>t \mapsto \min \{t, 1\},</math> and <math>t \mapsto \frac{t}{1 + t}.</math>{{sfn|Schechter|1996|p=487}} 
If <math>d</math> is a pseudometric (respectively, metric) on <math>X</math> and <math>R</math> is a bounded remetrization function then <math>R \circ d</math> is a bounded pseudometric (respectively, bounded metric) on <math>X</math> that is uniformly equivalent to <math>d.</math>{{sfn|Schechter|1996|p=487}} 

Suppose that <math>p_\bull = \left(p_i\right)_{i=1}^\infty</math> is a family of non-negative ''F''-seminorm on a vector space <math>X,</math> <math>R</math> is a bounded remetrization function, and <math>r_\bull = \left(r_i\right)_{i=1}^\infty</math> is a sequence of positive real numbers whose sum is finite. 
Then 
<math display="block">p(x) := \sum_{i=1}^\infty r_i R\left(p_i(x)\right)</math>
defines a bounded ''F''-seminorm that is uniformly equivalent to the <math>p_\bull.</math>{{sfn|Schechter|1996|pp=692-693}} 
It has the property that for any net <math>x_\bull = \left(x_a\right)_{a \in A}</math> in <math>X,</math> <math>p\left(x_\bull\right) \to 0</math> if and only if <math>p_i\left(x_\bull\right) \to 0</math> for all <math>i.</math>{{sfn|Schechter|1996|pp=692-693}} 
<math>p</math> is an ''F''-norm if and only if the <math>p_\bull</math> separate points on <math>X.</math>{{sfn|Schechter|1996|pp=692-693}}

==Characterizations==

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

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

===Of pseudometrizable TVS===

If <math>(X, \tau)</math> is a [[topological vector space]] (TVS) (where note in particular that <math>\tau</math> is assumed to be a vector topology) then the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=91-95}} 
<ol>
<li><math>X</math> is pseudometrizable (i.e. the vector topology <math>\tau</math> is induced by a pseudometric on <math>X</math>).</li>
<li><math>X</math> has a countable neighborhood base at the origin.</li>
<li>The topology on <math>X</math> is induced by a translation-invariant pseudometric on <math>X.</math></li>
<li>The topology on <math>X</math> is induced by an ''F''-seminorm.</li>
<li>The topology on <math>X</math> is induced by a paranorm.</li>
</ol>

===Of metrizable TVS===

If <math>(X, \tau)</math> is a TVS then the following are equivalent:
<ol>
<li><math>X</math> is metrizable.</li>
<li><math>X</math> is [[Hausdorff space|Hausdorff]] and pseudometrizable.</li>
<li><math>X</math> is Hausdorff and has a countable neighborhood base at the origin.{{sfn|Narici|Beckenstein|2011|pp=91-95}}{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>The topology on <math>X</math> is induced by a translation-invariant metric on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=91-95}}</li>
<li>The topology on <math>X</math> is induced by an ''F''-norm.{{sfn|Narici|Beckenstein|2011|pp=91-95}}{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>The topology on <math>X</math> is induced by a monotone ''F''-norm.{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>The topology on <math>X</math> is induced by a total paranorm.</li>
</ol>

{{Math theorem|name=[[Birkhoff–Kakutani theorem]]|math_statement=
If <math>(X, \tau)</math> is a topological vector space then the following three conditions are equivalent:<ref name="Kŏthe 1983">{{harvnb|Köthe|1983|loc=section 15.11}}</ref><ref group=note>In fact, this is true for topological group, for the proof doesn't use the scalar multiplications.</ref>
<ol>
<li>The origin <math>\{ 0 \}</math> is closed in <math>X,</math> and there is a [[countable]] [[neighborhood basis|basis of neighborhoods]] for <math>0</math> in <math>X.</math></li>
<li><math>(X, \tau)</math> is [[Metrizable space|metrizable]] (as a topological space).</li>
<li>There is a [[translation-invariant metric]] on <math>X</math> that induces on <math>X</math> the topology <math>\tau,</math> which is the given topology on <math>X.</math></li>
</ol>
By the Birkhoff–Kakutani theorem, it follows that there is an [[equivalence of metrics|equivalent metric]] that is translation-invariant.
}}

===Of locally convex pseudometrizable TVS===

If <math>(X, \tau)</math> is TVS then the following are equivalent:{{sfn|Narici|Beckenstein|2011|p=123}} 
<ol>
<li><math>X</math> is [[locally convex]] and pseudometrizable.</li>
<li><math>X</math> has a countable neighborhood base at the origin consisting of convex sets.</li>
<li>The topology of <math>X</math> is induced by a countable family of (continuous) seminorms.</li>
<li>The topology of <math>X</math> is induced by a countable increasing sequence of (continuous) seminorms <math>\left(p_i\right)_{i=1}^{\infty}</math> (increasing means that for all <math>i,</math> <math>p_i \geq p_{i+1}.</math></li>
<li>The topology of <math>X</math> is induced by an ''F''-seminorm of the form: 
<math display="block">p(x) = \sum_{n=1}^{\infty} 2^{-n} \operatorname{arctan} p_n(x)</math>
where <math>\left(p_i\right)_{i=1}^{\infty}</math> are (continuous) seminorms on <math>X.</math>{{sfn|Schechter|1996|p=706}}</li>
</ol>

==Quotients==

Let <math>M</math> be a vector subspace of a topological vector space <math>(X, \tau).</math> 
<ul>
<li>If <math>X</math> is a pseudometrizable TVS then so is <math>X / M.</math>{{sfn|Narici|Beckenstein|2011|pp=91-95}}</li>
<li>If <math>X</math> is a complete pseudometrizable TVS and <math>M</math> is a closed vector subspace of <math>X</math> then <math>X / M</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=91-95}}</li>
<li>If <math>X</math> is metrizable TVS and <math>M</math> is a closed vector subspace of <math>X</math> then <math>X / M</math> is metrizable.{{sfn|Narici|Beckenstein|2011|pp=91-95}}</li>
<li>If <math>p</math> is an ''F''-seminorm on <math>X,</math> then the map <math>P : X / M \to \R</math> defined by
<math display="block">P(x + M) := \inf_{} \{ p(x + m) : m \in M \}</math>
is an ''F''-seminorm on <math>X / M</math> that induces the usual [[quotient topology]] on <math>X / M.</math>{{sfn|Narici|Beckenstein|2011|pp=91-95}}  If in addition <math>p</math> is an ''F''-norm on <math>X</math> and if <math>M</math> is a closed vector subspace of <math>X</math> then <math>P</math> is an ''F''-norm on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=91-95}}</li>
</ul>

==Examples and sufficient conditions==

<ul>
<li>Every [[seminormed space]] <math>(X, p)</math> is pseudometrizable with a canonical pseudometric given by <math>d(x, y) := p(x - y)</math> for all <math>x, y \in X.</math>{{sfn|Narici|Beckenstein|2011|pp=115-154}}.</li>
<li>If <math>(X, d)</math> is pseudometric ''TVS'' with a translation invariant pseudometric <math>d,</math> then <math>p(x) := d(x, 0)</math> defines a paranorm.{{sfn|Wilansky|2013|pp=15-16}}  However, if <math>d</math> is a translation invariant pseudometric on the vector space <math>X</math> (without the addition condition that <math>(X, d)</math> is {{em|pseudometric TVS}}), then <math>d</math> need not be either an ''F''-seminorm{{sfn|Schaefer|Wolff|1999|pp=91-92}} nor a paranorm.</li>
<li>If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>Suppose <math>X</math> is either a [[DF-space]] or an [[LM-space]]. If <math>X</math> is a [[sequential space]] then it is either metrizable or else a [[Montel space|Montel]] DF-space.</li>
</ul>

If <math>X</math> is Hausdorff locally convex TVS then <math>X</math> with the [[Strong dual space|strong topology]], <math>\left(X, b\left(X, X^{\prime}\right)\right),</math> is metrizable if and only if there exists a countable set <math>\mathcal{B}</math> of bounded subsets of <math>X</math> such that every bounded subset of <math>X</math> is contained in some element of <math>\mathcal{B}.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}}

The [[strong dual space]] <math>X_b^{\prime}</math> of a metrizable locally convex space (such as a [[Fréchet space]]<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref>) <math>X</math> is a [[DF-space]].{{sfn|Schaefer|Wolff|1999|p=154}} 
The strong dual of a DF-space is a [[Fréchet space]].{{sfn|Schaefer|Wolff|1999|p=196}} 
The strong dual of a [[Reflexive space|reflexive]] Fréchet space is a [[bornological space]].{{sfn|Schaefer|Wolff|1999|p=154}} 
The strong bidual (that is, the [[strong dual space]] of the strong dual space) of a metrizable locally convex space is a Fréchet space.{{sfn|Schaefer|Wolff|1999|p=153}}
If <math>X</math> is a metrizable locally convex space then its strong dual <math>X_b^{\prime}</math> has one of the following properties, if and only if it has all of these properties: (1) [[Bornological space|bornological]], (2) [[Infrabarrelled space|infrabarreled]], (3) [[Barrelled space|barreled]].{{sfn|Schaefer|Wolff|1999|p=153}} 

===Normability===

A topological vector space is [[Seminormed space|seminormable]] if and only if it has a [[Convex set|convex]] bounded neighborhood of the origin. 
Moreover, a TVS is [[Normable space|normable]] if and only if it is [[Hausdorff space|Hausdorff]] and seminormable.{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
Every metrizable TVS on a finite-[[Hamel basis|dimensional]] vector space is a normable [[locally convex]] [[Complete topological vector space|complete TVS]], being [[TVS isomorphism|TVS-isomorphic]] to [[Euclidean space]]. Consequently, any metrizable TVS that is {{em|not}} normable must be infinite dimensional. 

If <math>M</math> is a metrizable [[Locally convex topological vector space|locally convex TVS]] that possess a [[Countable set|countable]] fundamental system of bounded sets, then <math>M</math> is normable.{{sfn|Schaefer|Wolff|1999|pp=68-72}} 

If <math>X</math> is a Hausdorff [[Locally convex topological vector space|locally convex space]] then the following are equivalent:

<ol>
<li><math>X</math> is [[Normable space|normable]].</li>
<li><math>X</math> has a [[Bounded set (topological vector space)|(von Neumann) bounded]] neighborhood of the origin.</li>
<li>the [[strong dual space]] <math>X^{\prime}_b</math> of <math>X</math> is normable.{{sfn|Trèves|2006|p=201}}</li>
</ol>

and if this locally convex space <math>X</math> is also metrizable, then the following may be appended to this list:

<ol start=4>
<li>the strong dual space of <math>X</math> is metrizable.{{sfn|Trèves|2006|p=201}}</li>
<li>the strong dual space of <math>X</math> is a [[Fréchet–Urysohn space|Fréchet–Urysohn]] locally convex space.<ref name="Gabriyelyan 2014"/></li>
</ol>

In particular, if a metrizable locally convex space <math>X</math> (such as a [[Fréchet space]]) is {{em|not}} normable then its [[strong dual space]] <math>X^{\prime}_b</math> is not a [[Fréchet–Urysohn space]] and consequently, this [[Complete topological vector space|complete]] Hausdorff locally convex space <math>X^{\prime}_b</math> is also neither metrizable nor normable.

Another consequence of this is that if <math>X</math> is a [[Reflexive space|reflexive]] [[locally convex]] TVS whose strong dual <math>X^{\prime}_b</math> is metrizable then <math>X^{\prime}_b</math> is necessarily a reflexive Fréchet space, <math>X</math> is a [[DF-space]], both <math>X</math> and <math>X^{\prime}_b</math> are necessarily [[Complete topological vector space|complete]] Hausdorff [[Ultrabornological space|ultrabornological]] [[Distinguished space|distinguished]] [[webbed space]]s, and moreover, <math>X^{\prime}_b</math> is normable if and only if <math>X</math> is normable if and only if <math>X</math> is Fréchet–Urysohn if and only if <math>X</math> is metrizable. In particular, such a space <math>X</math> is either a [[Banach space]] or else it is not even a Fréchet–Urysohn space.

==Metrically bounded sets and bounded sets==

Suppose that <math>(X, d)</math> is a pseudometric space and <math>B \subseteq X.</math> 
The set <math>B</math> is '''metrically bounded''' or '''<math>d</math>-bounded''' if there exists a real number <math>R > 0</math> such that <math>d(x, y) \leq R</math> for all <math>x, y \in B</math>; 
the smallest such <math>R</math> is then called the '''diameter''' or '''<math>d</math>-diameter''' of <math>B.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
If <math>B</math> is [[Bounded set (functional analysis)|bounded]] in a pseudometrizable TVS <math>X</math> then it is metrically bounded; 
the converse is in general false but it is true for [[locally convex]] metrizable TVSs.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

==Properties of pseudometrizable TVS==

{{Math theorem|name=Theorem{{sfn|Wilansky|2013|p=57}}|math_statement=
All infinite-dimensional [[Separable space|separable]] complete metrizable TVS are [[homeomorphic]].
}}

<ul>
<li>Every metrizable [[locally convex]] TVS is a [[quasibarrelled space]],{{sfn|Jarchow|1981|p=222}} [[bornological space]], and a [[Mackey space]].</li>
<li>Every complete {{em|pseudo}}metrizable TVS is a [[barrelled space]] and a [[Baire space]] (and hence non-meager).{{sfn|Narici|Beckenstein|2011|pp=371-423}}  However, there exist metrizable Baire spaces that are not [[Complete topological vector space|complete]].{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>If <math>X</math> is a metrizable locally convex space, then the strong dual of <math>X</math> is [[bornological space|bornological]] if and only if it is [[barreled space|barreled]], if and only if it is [[infrabarreled space|infrabarreled]].{{sfn|Schaefer|Wolff|1999|p=153}}</li>
<li>If <math>X</math> is a complete pseudometrizable TVS and <math>M</math> is a closed vector subspace of <math>X,</math> then <math>X / M</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=91-95}}</li>
<li>The [[strong dual space|strong dual]] of a locally convex metrizable TVS is a [[webbed space]].{{sfn|Narici|Beckenstein|2011|pp=459-483}}</li>
<li>If <math>(X, \tau)</math> and <math>(X, \nu)</math> are complete metrizable TVSs (i.e. [[F-space]]s) and if <math>\nu</math> is coarser than <math>\tau</math> then <math>\tau = \nu</math>;{{sfn|Köthe|1969|p=168}} this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete.{{sfn|Wilansky|2013|p=59}} Said differently, if <math>(X, \tau)</math> and <math>(X, \nu)</math> are both [[F-space]]s but with different topologies, then neither one of <math>\tau</math> and <math>\nu</math> contains the other as a subset. One particular consequence of this is, for example, that if <math>(X, p)</math> is a [[Banach space]] and <math>(X, q)</math> is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of <math>(X, p)</math> (i.e. if <math>p \leq C q</math> or if <math>q \leq C p</math> for some constant <math>C > 0</math>), then the only way that <math>(X, q)</math> can be a Banach space (i.e. also be complete) is if these two norms <math>p</math> and <math>q</math> are [[Equivalent norms|equivalent]]; if they are not equivalent, then <math>(X, q)</math> can not be a Banach space. 
As another consequence, if <math>(X, p)</math> is a Banach space and <math>(X, \nu)</math> is a [[Fréchet space]], then the map <math>p : (X, \nu) \to \R</math> is continuous if and only if the Fréchet space <math>(X, \nu)</math> {{em|is}} the TVS <math>(X, p)</math> (here, the Banach space <math>(X, p)</math> is being considered as a TVS, which means that its norm is "[[Forgetful functor|forgetten]]" but its topology is remembered). 
</li>
<li>A metrizable [[locally convex]] space is [[Normable space|normable]] if and only if its [[strong dual space]] is a [[Fréchet–Urysohn space|Fréchet–Urysohn]] locally convex space.<ref name="Gabriyelyan 2014"/></li>
<li>Any product of complete metrizable TVSs is a [[Baire space]].{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension <math>0.</math>{{sfn|Schaefer|Wolff|1999|pp=12-35}}</li>
<li>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.</li>
<li>Every complete {{em|pseudo}}metrizable TVS is a [[barrelled space]] and a [[Baire space]] (and thus non-meager).{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
<li>The dimension of a complete metrizable TVS is either finite or uncountable.{{sfn|Schaefer|Wolff|1999|pp=12-35}}</li>
</ul>

===Completeness===

{{Main|Complete topological vector space}}

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 <math>X</math> is a metrizable TVS and <math>d</math> is a metric that defines <math>X</math>'s topology, then its possible that <math>X</math> is complete as a TVS (i.e. relative to its uniformity) but the metric <math>d</math> is {{em|not}} a [[complete metric]] (such metrics exist even for <math>X = \R</math>). 
Thus, if <math>X</math> is a TVS whose topology is induced by a pseudometric <math>d,</math> then the notion of completeness of <math>X</math> (as a TVS) and the notion of completeness of the pseudometric space <math>(X, d)</math> are not always equivalent. 
The next theorem gives a condition for when they are equivalent: 

{{Math theorem|name=Theorem|math_statement=
If <math>X</math> is a pseudometrizable TVS whose topology is induced by a {{em|translation invariant}} pseudometric <math>d,</math> then <math>d</math> is a complete pseudometric on <math>X</math> if and only if <math>X</math> is complete as a TVS.{{sfn|Narici|Beckenstein|2011|pp=47-50}} 
}}

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|p=35}}<ref name="Klee Inv metrics">{{Cite journal|last1=Klee|first1=V. L.|title=Invariant metrics in groups (solution of a problem of Banach)|year=1952|journal=Proc. Amer. Math. Soc.|volume=3|issue=3|pages=484–487|url=https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf|doi = 10.1090/s0002-9939-1952-0047250-4 |doi-access=free}}</ref>|note=Klee|math_statement=
Let <math>d</math> be {{em|any}}<ref group=note>Not assumed to be translation-invariant.</ref> metric on a vector space <math>X</math> such that the topology <math>\tau</math> induced by <math>d</math> on <math>X</math> makes <math>(X, \tau)</math> into a topological vector space. If <math>(X, d)</math> is a complete metric space then <math>(X, \tau)</math> is a complete-TVS.
}}

{{Math theorem|name=Theorem|math_statement=
If <math>X</math> is a TVS whose topology is induced by a paranorm <math>p,</math> then <math>X</math> is complete if and only if for every sequence <math>\left(x_i\right)_{i=1}^{\infty}</math> in <math>X,</math> if <math>\sum_{i=1}^{\infty} p\left(x_i\right) < \infty</math> then <math>\sum_{i=1}^{\infty} x_i</math> converges in <math>X.</math>{{sfn|Wilansky|2013|pp=56-57}}  
}}

If <math>M</math> is a closed vector subspace of a complete pseudometrizable TVS <math>X,</math> then the quotient space <math>X / M</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=47-66}}  
If <math>M</math> is a {{em|complete}} vector subspace of a metrizable TVS <math>X</math> and if the quotient space <math>X / M</math> is complete then so is <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}  If <math>X</math> is not complete then <math>M := X,</math> but not complete, vector subspace of <math>X.</math>  

A [[Baire space|Baire]] [[Separable space|separable]] [[topological group]] is metrizable if and only if it is cosmic.<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref>

===Subsets and subsequences===

<ul>
<li>Let <math>M</math> be a [[Separable space|separable]] locally convex metrizable topological vector space and let <math>C</math> be its completion. If <math>S</math> is a bounded subset of <math>C</math> then there exists a bounded subset <math>R</math> of <math>X</math> such that <math>S \subseteq \operatorname{cl}_C R.</math>{{sfn|Schaefer|Wolff|1999|pp=190-202}}</li>
<li>Every [[Totally bounded set|totally bounded]] subset of a locally convex metrizable TVS <math>X</math> is contained in the closed [[Absolutely convex set|convex balanced hull]] of some sequence in <math>X</math> that converges to <math>0.</math></li>
<li>In a pseudometrizable TVS, every [[bornivore]] is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=172-173}}</li>
<li>If <math>d</math> is a translation invariant metric on a vector space <math>X,</math> then <math>d(n x, 0) \leq n d(x, 0)</math> for all <math>x \in X</math> and every positive integer <math>n.</math>{{sfn|Rudin|1991|p=22}}</li>
<li>If <math>\left(x_i\right)_{i=1}^{\infty}</math> is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence <math>\left(r_i\right)_{i=1}^{\infty}</math> of positive real numbers diverging to <math>\infty</math> such that <math>\left(r_i x_i\right)_{i=1}^{\infty} \to 0.</math>{{sfn|Rudin|1991|p=22}}</li>
<li>A subset of a complete metric space is closed if and only if it is complete.  If a space <math>X</math> is not complete, then <math>X</math> is a closed subset of <math>X</math> that is not complete.</li>
<li>If <math>X</math> is a metrizable [[Locally convex topological vector space|locally convex]] TVS then for every [[Bounded set (topological vector space)|bounded]] subset <math>B</math> of <math>X,</math> there exists a bounded [[Absolutely convex set|disk]] <math>D</math> in <math>X</math> such that <math>B \subseteq X_D,</math> and both <math>X</math> and the [[auxiliary normed space]] <math>X_D</math> induce the same [[subspace topology]] on <math>B.</math>{{sfn|Narici|Beckenstein|2011|pp=441-457}}</li>
</ul>

{{Math theorem|name=[[Banach-Saks theorem]]{{sfn|Rudin|1991|p=67}}|math_statement=
If <math>\left(x_n\right)_{n=1}^{\infty}</math> is a sequence in a [[locally convex]] metrizable TVS <math>(X, \tau)</math> that converges {{em|weakly}} to some <math>x \in X,</math> then there exists a sequence <math>y_{\bull} = \left(y_i\right)_{i=1}^{\infty}</math> in <math>X</math> such that <math>y_{\bull} \to x</math> in <math>(X, \tau)</math> and each <math>y_i</math> is a convex combination of finitely many <math>x_n.</math> 
}}

{{Math theorem|name=Mackey's countability condition{{sfn|Narici|Beckenstein|2011|pp=156-175}}|math_statement=
Suppose that <math>X</math> is a locally convex metrizable TVS and that <math>\left(B_i\right)_{i=1}^{\infty}</math> is a countable sequence of bounded subsets of <math>X.</math> 
Then there exists a bounded subset <math>B</math> of <math>X</math> and a sequence <math>\left(r_i\right)_{i=1}^{\infty}</math> of positive real numbers such that <math>B_i \subseteq r_i B</math> for all <math>i.</math>
}}

'''Generalized series'''

As described [[Series (mathematics)#Summations over arbitrary index sets|in this article's section on generalized series]], for any <math>I</math>-[[Indexing set|indexed family]] family <math>\left(r_i\right)_{i \in I}</math> of vectors from a TVS <math>X,</math> it is possible to define their sum <math>\textstyle\sum\limits_{i \in I} r_i</math> as the limit of the [[Net (mathematics)|net]] of finite partial sums <math>F \in \operatorname{FiniteSubsets}(I) \mapsto \textstyle\sum\limits_{i \in F} r_i</math> where the domain <math>\operatorname{FiniteSubsets}(I)</math> is [[Directed set|directed]] by <math>\,\subseteq.\,</math> 
If <math>I = \N</math> and <math>X = \Reals,</math> for instance, then the generalized series <math>\textstyle\sum\limits_{i \in \N} r_i</math> converges if and only if <math>\textstyle\sum\limits_{i=1}^\infty r_i</math> [[Unconditional convergence|converges unconditionally]] in the usual sense (which for real numbers, [[Riemann series theorem|is equivalent]] to [[absolute convergence]]). 
If a generalized series <math>\textstyle\sum\limits_{i \in I} r_i</math> converges in a metrizable TVS, then the set <math>\left\{i \in I : r_i \neq 0\right\}</math> is necessarily [[Countable set|countable]] (that is, either finite or [[countably infinite]]);<ref group=proof name=ProofCountablyManyNon0Terms /> 
in other words, all but at most countably many <math>r_i</math> will be zero and so this generalized series <math>\textstyle\sum\limits_{i \in I} r_i ~=~ \textstyle\sum\limits_{\stackrel{i \in I}{r_i \neq 0}} r_i</math> is actually a sum of at most countably many non-zero terms.

===Linear maps===

If <math>X</math> is a pseudometrizable TVS and <math>A</math> maps bounded subsets of <math>X</math> to bounded subsets of <math>Y,</math> then <math>A</math> is continuous.{{sfn|Narici|Beckenstein|2011|pp=156-175}}  
Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.{{sfn|Narici|Beckenstein|2011|p=125}}  Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its [[algebraic dual space]].{{sfn|Narici|Beckenstein|2011|p=125}}  

If <math>F : X \to Y</math> is a linear map between TVSs and <math>X</math> is metrizable then the following are equivalent:

<ol>
<li><math>F</math> is continuous;</li>
<li><math>F</math> is a (locally) bounded map (that is, <math>F</math> maps [[Bounded set (topological vector space)|(von Neumann) bounded subsets]] of <math>X</math> to bounded subsets of <math>Y</math>);{{sfn|Jarchow|1981|pp=38-42}}</li>
<li><math>F</math> is [[sequentially continuous]];{{sfn|Jarchow|1981|pp=38-42}}</li>
<li>the image under <math>F</math> of every null sequence in <math>X</math> is a bounded set{{sfn|Jarchow|1981|pp=38-42}} where by definition, a '''{{em|null sequence}}''' is a sequence that converges to the origin.</li>
<li><math>F</math> maps null sequences to null sequences;</li>
</ol>

'''Open and almost open maps'''

:'''Theorem''': If <math>X</math> is a complete pseudometrizable TVS, <math>Y</math> is a Hausdorff TVS, and <math>T : X \to Y</math> is a closed and [[Almost open linear map|almost]] open linear surjection, then <math>T</math> is an open map.{{sfn|Narici|Beckenstein|2011|pp=466-468}}  

:'''Theorem''': If <math>T : X \to Y</math> is a surjective linear operator from a [[locally convex]] space <math>X</math> onto a [[barrelled space]] <math>Y</math> (e.g. every complete pseudometrizable space is barrelled) then <math>T</math> is [[Almost open linear map|almost open]].{{sfn|Narici|Beckenstein|2011|pp=466-468}}  

:'''Theorem''': If <math>T : X \to Y</math> is a surjective linear operator from a TVS <math>X</math> onto a [[Baire space]] <math>Y</math> then <math>T</math> is almost open.{{sfn|Narici|Beckenstein|2011|pp=466-468}}  

:'''Theorem''': Suppose <math>T : X \to Y</math> is a continuous linear operator from a complete pseudometrizable TVS <math>X</math> into a Hausdorff TVS <math>Y.</math> If the image of <math>T</math> is non-[[Meager set|meager]] in <math>Y</math> then <math>T : X \to Y</math> is a surjective open map and <math>Y</math> is a [[Complete topological vector space|complete]] metrizable space.{{sfn|Narici|Beckenstein|2011|pp=466-468}}  

===Hahn-Banach extension property===
{{Main|Hahn-Banach theorem}}

A vector subspace <math>M</math> of a TVS <math>X</math> has '''the extension property''' if any continuous linear functional on <math>M</math> can be extended to a continuous linear functional on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
Say that a TVS <math>X</math> has the '''[[Hahn-Banach theorem|Hahn-Banach]] extension property''' ('''HBEP''') if every vector subspace of <math>X</math> has the extension property.{{sfn|Narici|Beckenstein|2011|pp=225-273}} 

The [[Hahn-Banach theorem]] guarantees that every Hausdorff locally convex space has the HBEP. 
For complete metrizable TVSs there is a converse:

{{Math theorem|name=Theorem|note=Kalton|math_statement=
Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.{{sfn|Narici|Beckenstein|2011|pp=225-273}}  
}}

If a vector space <math>X</math> 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.{{sfn|Narici|Beckenstein|2011|pp=225-273}}

==See also==

* {{annotated link|Asymmetric norm}}
* {{annotated link|Complete metric space}}
* {{annotated link|Complete topological vector space}}
* {{annotated link|Equivalence of metrics}}
* {{annotated link|F-space}}
* {{annotated link|Fréchet space}}
* {{annotated link|Generalised metric}}
* {{annotated link|K-space (functional analysis)}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Metric space}}
* {{annotated link|Pseudometric space}}
* {{annotated link|Relation of norms and metrics}}
* {{annotated link|Seminorm}}
* {{annotated link|Sublinear function}}
* {{annotated link|Uniform space}}
* {{annotated link|Ursescu theorem}}

==Notes==

{{reflist|group=note}}

'''Proofs'''

{{reflist|group=proof|refs=
<ref name=ProofCountablyManyNon0Terms>Suppose the net <math display=inline>\textstyle\sum\limits_{i \in I} r_i ~\stackrel{\scriptscriptstyle\text{def}}{=}~ {\textstyle\lim\limits_{A \in \operatorname{FiniteSubsets}(I)}} \ \textstyle\sum\limits_{i \in A} r_i = \lim \left\{\textstyle\sum\limits_{i\in A} r_i \,: A \subseteq I, A \text{ finite }\right\}</math> converges to some point in a metrizable TVS <math>X,</math> where recall that this net's domain is the [[directed set]] <math>(\operatorname{FiniteSubsets}(I), \subseteq).</math> 
Like every convergent net, this convergent net of partial sums <math>A \mapsto \textstyle\sum\limits_{i \in A} r_i</math> is a {{em|[[Cauchy net]]}}, which for this particular net means (by definition) that for every neighborhood <math>W</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that 
<math display=inline>\textstyle\sum\limits_{i \in B} r_i - \textstyle\sum\limits_{i \in C} r_i \in W</math> for all finite supersets <math>B, C \supseteq A_0;</math>
this implies that <math>r_i \in W</math> for every <math>i \in I \setminus A_0</math> (by taking <math>B := A_0 \cup \{i\}</math> and <math>C := A_0</math>). 
Since <math>X</math> is metrizable, it has a countable neighborhood basis <math>U_1, U_2, \ldots</math> at the origin, whose intersection is necessarily <math>U_1 \cap U_2 \cap \cdots = \{0\}</math> (since <math>X</math> is a Hausdorff TVS).
For every positive integer <math>n \in \N,</math> pick a finite subset <math>A_n \subseteq I</math> such that <math>r_i \in U_n</math> for every <math>i \in I \setminus A_n.</math> 
If <math>i</math> belongs to <math>(I \setminus A_1) \cap (I \setminus A_2) \cap \cdots = I \setminus \left(A_1 \cup A_2 \cup \cdots\right)</math> then <math>r_i</math> belongs to <math>U_1 \cap U_2 \cap \cdots = \{0\}.</math> 
Thus <math>r_i = 0</math> for every index <math>i \in I</math> that does not belong to the countable set <math>A_1 \cup A_2 \cup \cdots.</math> <math>\blacksquare</math></ref>
}}

==References==

{{reflist}}

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{{Functional analysis}}
{{Topological vector spaces}}

[[Category:Metric spaces]]
[[Category:Topological vector spaces]]