Metrizable topological vector space#Additive sequences

A pseudometric on a set $X$ is a map $d\; :\; X\; \backslash times\; X\; \backslash rarr\; \backslash R$ satisfying the following properties:

1. $d(x,\; x)\; =\; 0\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X$;
2. Symmetry: $d(x,\; y)\; =\; d(y,\; x)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X$;
3. Subadditivity: $d(x,\; z)\; \backslash leq\; d(x,\; y)\; +\; d(y,\; z)\; \backslash text\{\; for\; all\; \}\; x,\; y,\; z\; \backslash in\; X.$

A pseudometric is called a metric if it satisfies:

4. Identity of indiscernibles: for all $x,\; y\; \backslash in\; X,$ if $d(x,\; y)\; =\; 0$ then $x\; =\; y.$

Ultrapseudometric

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

5. Strong/Ultrametric triangle inequality: $d(x,\; z)\; \backslash leq\; \backslash max\; \backslash \{\; d(x,\; y),\; d(y,\; z)\; \backslash \}\; \backslash text\{\; for\; all\; \}\; x,\; y,\; z\; \backslash in\; X.$

Pseudometric space

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

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

as $z$ ranges over $X$ and $r\; >\; 0$ ranges over the positive real numbers,

forms a basis for a topology on $X$ that is called the $d$-topology or the pseudometric topology on $X$ induced by $d.$

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

Pseudometrizable space

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

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

Every topological vector space (TVS) $X$ is an additive commutative topological group but not all group topologies on $X$ are vector topologies.

This is because despite it making addition and negation continuous, a group topology on a vector space $X$ may fail to make scalar multiplication continuous.

For instance, the discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

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

1. Translation invariance: $d(x\; +\; z,\; y\; +\; z)\; =\; d(x,\; y)\; \backslash text\{\; for\; all\; \}\; x,\; y,\; z\; \backslash in\; X$;
2. $d(x,\; y)\; =\; d(x\; -\; y,\; 0)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X.$

If $X$ is a topological group the a value or G-seminorm on $X$ (the G stands for Group) is a real-valued map $p\; :\; X\; \backslash rarr\; \backslash R$ with the following properties:{{sfn|Narici|Beckenstein|2011|pp=37-40}}

1. Non-negative: $p\; \backslash geq\; 0.$
2. Subadditive: $p(x\; +\; y)\; \backslash leq\; p(x)\; +\; p(y)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X$;
3. $p(0)\; =\; 0..$
4. Symmetric: $p(-x)\; =\; p(x)\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X.$

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

5. Total/Positive definite: If $p(x)\; =\; 0$ then $x\; =\; 0.$

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

• $|p(x)\; -\; p(y)|\; \backslash leq\; p(x\; -\; y)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X.${{sfn|Swartz|1992|p=15}}
• $p(n\; x)\; \backslash leq\; n\; p(x)$ and $\backslash frac\{1\}\{n\}\; p(x)\; \backslash leq\; p(x\; /\; n)$ for all $x\; \backslash in\; X$ and positive integers $n.${{sfn|Wilansky|2013|p=17}}
• The set $\backslash \{\; x\; \backslash in\; X\; :\; p(x)\; =\; 0\; \backslash \}$ is an additive subgroup of $X.${{sfn|Swartz|1992|p=15}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=37-40}}|math_statement=

Suppose that $X$ is an additive commutative group.

If $d$ is a translation invariant pseudometric on $X$ then the map $p(x)\; :=\; d(x,\; 0)$ is a value on $X$ called the value associated with $d$, and moreover, $d$ generates a group topology on $X$ (i.e. the $d$-topology on $X$ makes $X$ into a topological group).

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

}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=37-40}}|math_statement=

If $(X,\; \backslash tau)$ is an additive commutative topological group then the following are equivalent:

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

If $(X,\; \backslash tau)$ 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.

}}

Let $X$ be a non-trivial (i.e. $X\; \backslash neq\; \backslash \{\; 0\; \backslash \}$) real or complex vector space and let $d$ be the translation-invariant trivial metric on $X$ defined by $d(x,\; x)\; =\; 0$ and $d(x,\; y)\; =\; 1\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X$ such that $x\; \backslash neq\; y.$

The topology $\backslash tau$ that $d$ induces on $X$ is the discrete topology, which makes $(X,\; \backslash tau)$ into a commutative topological group under addition but does {{em|not}} form a vector topology on $X$ because $(X,\; \backslash tau)$ is disconnected but every vector topology is connected.

What fails is that scalar multiplication isn't continuous on $(X,\; \backslash tau).$

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.

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

{{Math theorem|name=Continuity of addition at 0|math_statement=

If $(X,\; +)$ is a group (as all vector spaces are), $\backslash tau$ is a topology on $X,$ and $X\; \backslash times\; X$ is endowed with the product topology, then the addition map $X\; \backslash times\; X\; \backslash to\; X$ (i.e. the map $(x,\; y)\; \backslash mapsto\; x\; +\; y$) is continuous at the origin of $X\; \backslash times\; X$ if and only if the set of neighborhoods of the origin in $(X,\; \backslash tau)$ 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 groups.

{{Math theorem|name=Theorem|math_statement=

Let $U\_\{\backslash bull\}\; =\; \backslash left(U\_i\backslash right)\_\{i=0\}^\{\backslash infty\}$ be a collection of subsets of a vector space such that $0\; \backslash in\; U\_i$ and $U\_\{i+1\}\; +\; U\_\{i+1\}\; \backslash subseteq\; U\_i$ for all $i\; \backslash geq\; 0.$

For all $u\; \backslash in\; U\_0,$ let

$$\backslash mathbb\{S\}(u)\; :=\; \backslash left\backslash \{\; n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)\; ~:~\; k\; \backslash geq\; 1,\; n\_i\; \backslash geq\; 0\; \backslash text\{\; for\; all\; \}\; i,\; \backslash text\{\; and\; \}\; u\; \backslash in\; U\_\{n\_1\}\; +\; \backslash cdots\; +\; U\_\{n\_k\}\backslash right\backslash \}.$$

Define $f\; :\; X\; \backslash to\; [0,\; 1]$ by $f(x)\; =\; 1$ if $x\; \backslash not\backslash in\; U\_0$ and otherwise let

$$f(x)\; :=\; \backslash inf\_\{\}\; \backslash left\backslash \{\; 2^\{-\; n\_1\}\; +\; \backslash cdots\; 2^\{-\; n\_k\}\; ~:~\; n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)\; \backslash in\; \backslash mathbb\{S\}(x)\backslash right\backslash \}.$$

Then $f$ is subadditive (meaning $f(x\; +\; y)\; \backslash leq\; f(x)\; +\; f(y)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X$) and $f\; =\; 0$ on $\backslash bigcap\_\{i\; \backslash geq\; 0\}\; U\_i,$ so in particular $f(0)\; =\; 0.$

If all $U\_i$ are symmetric sets then $f(-x)\; =\; f(x)$ and if all $U\_i$ are balanced then $f(s\; x)\; \backslash leq\; f(x)$ for all scalars $s$ such that $|s|\; \backslash leq\; 1$ and all $x\; \backslash in\; X.$

If $X$ is a topological vector space and if all $U\_i$ are neighborhoods of the origin then $f$ is continuous, where if in addition $X$ is Hausdorff and $U\_\{\backslash bull\}$ forms a basis of balanced neighborhoods of the origin in $X$ then $d(x,\; y)\; :=\; f(x\; -\; y)$ is a metric defining the vector topology on $X.$

}}

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

Assume that $n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)$ always denotes a finite sequence of non-negative integers and use the notation:

$$\backslash sum\; 2^\{-\; n\_\{\backslash bull\}\}\; :=\; 2^\{-\; n\_1\}\; +\; \backslash cdots\; +\; 2^\{-\; n\_k\}\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; \backslash sum\; U\_\{n\_\{\backslash bull\}\}\; :=\; U\_\{n\_1\}\; +\; \backslash cdots\; +\; U\_\{n\_k\}.$$

For any integers $n\; \backslash geq\; 0$ and $d\; >\; 2,$

$$U\_n\; \backslash supseteq\; U\_\{n+1\}\; +\; U\_\{n+1\}\; \backslash supseteq\; U\_\{n+1\}\; +\; U\_\{n+2\}\; +\; U\_\{n+2\}\; \backslash supseteq\; U\_\{n+1\}\; +\; U\_\{n+2\}\; +\; \backslash cdots\; +\; U\_\{n+d\}\; +\; U\_\{n+d+1\}\; +\; U\_\{n+d+1\}.$$

From this it follows that if $n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)$ consists of distinct positive integers then $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; U\_\{-1\; +\; \backslash min\; \backslash left(n\_\{\backslash bull\}\backslash right)\}.$

It will now be shown by induction on $k$ that if $n\_\{\backslash bull\}\; =\; \backslash left(n\_1,\; \backslash ldots,\; n\_k\backslash right)$ consists of non-negative integers such that $\backslash sum\; 2^\{-\; n\_\{\backslash bull\}\}\; \backslash leq\; 2^\{-\; M\}$ for some integer $M\; \backslash geq\; 0$ then $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; U\_M.$

This is clearly true for $k\; =\; 1$ and $k\; =\; 2$ so assume that $k\; >\; 2,$ which implies that all $n\_i$ are positive.

If all $n\_i$ are distinct then this step is done, and otherwise pick distinct indices such that $n\_i\; =\; n\_j$ and construct $m\_\{\backslash bull\}\; =\; \backslash left(m\_1,\; \backslash ldots,\; m\_\{k-1\}\backslash right)$ from $n\_\{\backslash bull\}$ by replacing each $n\_i$ with $n\_i\; -\; 1$ and deleting the $j^\{\backslash text\{th\}\}$ element of $n\_\{\backslash bull\}$ (all other elements of $n\_\{\backslash bull\}$ are transferred to $m\_\{\backslash bull\}$ unchanged).

Observe that $\backslash sum\; 2^\{-\; n\_\{\backslash bull\}\}\; =\; \backslash sum\; 2^\{-\; m\_\{\backslash bull\}\}$ and $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; \backslash sum\; U\_\{m\_\{\backslash bull\}\}$ (because $U\_\{n\_i\}\; +\; U\_\{n\_j\}\; \backslash subseteq\; U\_\{n\_i\; -\; 1\}$) so by appealing to the inductive hypothesis we conclude that $\backslash sum\; U\_\{n\_\{\backslash bull\}\}\; \backslash subseteq\; \backslash sum\; U\_\{m\_\{\backslash bull\}\}\; \backslash subseteq\; U\_M,$ as desired.

It is clear that $f(0)\; =\; 0$ and that $0\; \backslash leq\; f\; \backslash leq\; 1$ so to prove that $f$ is subadditive, it suffices to prove that $f(x\; +\; y)\; \backslash leq\; f(x)\; +\; f(y)$ when $x,\; y\; \backslash in\; X$ are such that which implies that $x,\; y\; \backslash in\; U\_0.$

This is an exercise.

If all $U\_i$ are symmetric then $x\; \backslash in\; \backslash sum\; U\_\{n\_\{\backslash bull\}\}$ if and only if $-\; x\; \backslash in\; \backslash sum\; U\_\{n\_\{\backslash bull\}\}$ from which it follows that $f(-x)\; \backslash leq\; f(x)$ and $f(-x)\; \backslash geq\; f(x).$

If all $U\_i$ are balanced then the inequality $f(s\; x)\; \backslash leq\; f(x)$ for all unit scalars $s$ such that $|s|\; \backslash leq\; 1$ is proved similarly.

Because $f$ is a nonnegative subadditive function satisfying $f(0)\; =\; 0,$ as described in the article on sublinear functionals, $f$ is uniformly continuous on $X$ if and only if $f$ is continuous at the origin.

If all $U\_i$ are neighborhoods of the origin then for any real $r\; >\; 0,$ pick an integer $M\; >\; 1$ such that so that $x\; \backslash in\; U\_M$ implies

If the set of all $U\_i$ form basis of balanced neighborhoods of the origin then it may be shown that for any $n\; >\; 1,$ there exists some such that implies $x\; \backslash in\; U\_n.$ $\backslash blacksquare$

{{collapse bottom}}

If $X$ is a vector space over the real or complex numbers then a paranorm on $X$ is a G-seminorm (defined above) $p\; :\; X\; \backslash rarr\; \backslash R$ on $X$ that satisfies any of the following additional conditions, each of which begins with "for all sequences $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X$ and all convergent sequences of scalars $s\_\{\backslash bull\}\; =\; \backslash left(s\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$":{{sfn|Wilansky|2013|p=15}}

1. Continuity of multiplication: if $s$ is a scalar and $x\; \backslash in\; X$ are such that $p\backslash left(x\_i\; -\; x\backslash right)\; \backslash to\; 0$ and $s\_\{\backslash bull\}\; \backslash to\; s,$ then $p\backslash left(s\_i\; x\_i\; -\; s\; x\backslash right)\; \backslash to\; 0.$
2. Both of the conditions:
   • if $s\_\{\backslash bull\}\; \backslash to\; 0$ and if $x\; \backslash in\; X$ is such that $p\backslash left(x\_i\; -\; x\backslash right)\; \backslash to\; 0$ then $p\backslash left(s\_i\; x\_i\backslash right)\; \backslash to\; 0$;
   • if $p\backslash left(x\_\{\backslash bull\}\backslash right)\; \backslash to\; 0$ then $p\backslash left(s\; x\_i\backslash right)\; \backslash to\; 0$ for every scalar $s.$

3. Both of the conditions:
   • if $p\backslash left(x\_\{\backslash bull\}\backslash right)\; \backslash to\; 0$ and $s\_\{\backslash bull\}\; \backslash to\; s$ for some scalar $s$ then $p\backslash left(s\_i\; x\_i\backslash right)\; \backslash to\; 0$;
   • if $s\_\{\backslash bull\}\; \backslash to\; 0$ then $p\backslash left(s\_i\; x\backslash right)\; \backslash to\; 0\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X.$

4. Separate continuity:{{sfn|Schechter|1996|pp=689-691}}
   • if $s\_\{\backslash bull\}\; \backslash to\; s$ for some scalar $s$ then $p\backslash left(s\; x\_i\; -\; s\; x\backslash right)\; \backslash to\; 0$ for every $x\; \backslash in\; X$;
   • if $s$ is a scalar, $x\; \backslash in\; X,$ and $p\backslash left(x\_i\; -\; x\backslash right)\; \backslash to\; 0$ then $p\backslash left(s\; x\_i\; -\; s\; x\backslash right)\; \backslash to\; 0$ .

A paranorm is called total if in addition it satisfies:

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

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

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

• the set $\backslash \{\; x\; \backslash in\; X\; :\; p(x)\; =\; 0\; \backslash \}$ is a vector subspace of $X.${{sfn|Wilansky|2013|pp=15-18}}
• $p(x\; +\; n)\; =\; p(x)\; \backslash text\{\; for\; all\; \}\; x,\; n\; \backslash in\; X$ with $p(n)\; =\; 0.${{sfn|Wilansky|2013|pp=15-18}}
• If a paranorm $p$ satisfies $p(s\; x)\; \backslash leq\; |s|\; p(x)\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X$ and scalars $s,$ then $p$ is absolutely homogeneity (i.e. equality holds){{sfn|Wilansky|2013|pp=15-18}} and thus $p$ is a seminorm.

• If $d$ is a translation-invariant pseudometric on a vector space $X$ that induces a vector topology $\backslash tau$ on $X$ (i.e. $(X,\; \backslash tau)$ is a TVS) then the map $p(x)\; :=\; d(x\; -\; y,\; 0)$ defines a continuous paranorm on $(X,\; \backslash tau)$; moreover, the topology that this paranorm $p$ defines in $X$ is $\backslash tau.${{sfn|Wilansky|2013|pp=15-18}}
• If $p$ is a paranorm on $X$ then so is the map $q(x)\; :=\; p(x)\; /\; [1\; +\; p(x)].${{sfn|Wilansky|2013|pp=15-18}}
• Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
• Every seminorm is a paranorm.{{sfn|Wilansky|2013|pp=15-18}}
• The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).{{sfn|Schechter|1996|p=692}}
• The sum of two paranorms is a paranorm.{{sfn|Wilansky|2013|pp=15-18}}
• If $p$ and $q$ are paranorms on $X$ then so is $(p\; \backslash wedge\; q)(x)\; :=\; \backslash inf\_\{\}\; \backslash \{\; p(y)\; +\; q(z)\; :\; x\; =\; y\; +\; z\; \backslash text\{\; with\; \}\; y,\; z\; \backslash in\; X\; \backslash \}.$ Moreover, $(p\; \backslash wedge\; q)\; \backslash leq\; p$ and $(p\; \backslash wedge\; q)\; \backslash leq\; q.$ This makes the set of paranorms on $X$ into a conditionally complete lattice.{{sfn|Wilansky|2013|pp=15-18}}
• Each of the following real-valued maps are paranorms on $X\; :=\; \backslash R^2$:
  • $(x,\; y)\; \backslash mapsto\; |x|$
  • $(x,\; y)\; \backslash mapsto\; |x|\; +\; |y|$

• The real-valued maps $(x,\; y)\; \backslash mapsto\; \backslash sqrt\{\backslash left|x^2\; -\; y^2\backslash right|\}$ and $(x,\; y)\; \backslash mapsto\; \backslash left|x^2\; -\; y^2\backslash right|^\{3/2\}$ are {{em|not}} paranorms on $X\; :=\; \backslash R^2.${{sfn|Wilansky|2013|pp=15-18}}
• If $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a Hamel basis on a vector space $X$ then the real-valued map that sends $x\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; s\_i\; x\_i\; \backslash in\; X$ (where all but finitely many of the scalars $s\_i$ are 0) to $\backslash sum\_\{i\; \backslash in\; I\}\; \backslash sqrt\{\backslash left|s\_i\backslash right|\}$ is a paranorm on $X,$ which satisfies $p(sx)\; =\; \backslash sqrt$

  s

  p(x) for all $x\; \backslash in\; X$ and scalars $s.${{sfn|Wilansky|2013|pp=15-18}}
• The function $p(x)\; :=\; |\backslash sin\; (\backslash pi\; x)|\; +\; \backslash min\; \backslash \{\; 2,\; |x|\; \backslash \}$ is a paranorm on $\backslash R$ that is {{em|not}} balanced but nevertheless equivalent to the usual norm on $R.$ Note that the function $x\; \backslash mapsto\; |\backslash sin\; (\backslash pi\; x)|$ is subadditive.{{sfn|Schechter|1996|p=691}}
• Let $X\_\{\backslash Complex\}$ be a complex vector space and let $X\_\{\backslash R\}$ denote $X\_\{\backslash Complex\}$ considered as a vector space over $\backslash R.$ Any paranorm on $X\_\{\backslash Complex\}$ is also a paranorm on $X\_\{\backslash R\}.${{sfn|Schechter|1996|p=692}}

If $X$ is a vector space over the real or complex numbers then an F-seminorm on $X$ (the $F$ stands for Fréchet) is a real-valued map $p\; :\; X\; \backslash to\; \backslash Reals$ with the following four properties: {{sfn|Narici|Beckenstein|2011|pp=91-95}}

1. Non-negative: $p\; \backslash geq\; 0.$
2. Subadditive: $p(x\; +\; y)\; \backslash leq\; p(x)\; +\; p(y)$ for all $x,\; y\; \backslash in\; X$
3. Balanced: $p(a\; x)\; \backslash leq\; p(x)$ for $x\; \backslash in\; X$ all scalars $a$ satisfying $|a|\; \backslash leq\; 1;$
   • This condition guarantees that each set of the form $\backslash \{z\; \backslash in\; X\; :\; p(z)\; \backslash leq\; r\backslash \}$ or for some $r\; \backslash geq\; 0$ is a balanced set.

4. For every $x\; \backslash in\; X,$ $p\backslash left(\backslash tfrac\{1\}\{n\}\; x\backslash right)\; \backslash to\; 0$ as $n\; \backslash to\; \backslash infty$
   • The sequence $\backslash left(\backslash tfrac\{1\}\{n\}\backslash right)\_\{n=1\}^\backslash infty$ can be replaced by any positive sequence converging to the zero.{{sfn|Jarchow|1981|pp=38-42}}

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

5. Total/Positive definite: $p(x)\; =\; 0$ implies $x\; =\; 0.$

An F-seminorm is called monotone if it satisfies:

6. Monotone: for all non-zero $x\; \backslash in\; X$ and all real $s$ and $t$ such that {{sfn|Jarchow|1981|pp=38-42}}

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

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

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}}

• Every positive scalar multiple of an F-seminorm (resp. F-norm, seminorm) is again an F-seminorm (resp. F-norm, seminorm).
• The sum of finitely many F-seminorms (resp. F-norms) is an F-seminorm (resp. F-norm).
• If $p$ and $q$ are F-seminorms on $X$ then so is their pointwise supremum $x\; \backslash mapsto\; \backslash sup\; \backslash \{p(x),\; q(x)\backslash \}.$ The same is true of the supremum of any non-empty finite family of F-seminorms on $X.${{sfn|Jarchow|1981|pp=38-42}}
• 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}}
• A non-negative real-valued function on $X$ is a seminorm if and only if it is a convex F-seminorm, or equivalently, if and only if it is a convex balanced G-seminorm.{{sfn|Schechter|1996|p=691}} In particular, every seminorm is an F-seminorm.
• For any the map $f$ on $\backslash Reals^n$ defined by

  $$[f\backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right)]^p\; =\; \backslash left|x\_1\backslash right|^p\; +\; \backslash cdots\; \backslash left|x\_n\backslash right|^p$$

  is an F-norm that is not a norm.

• If $L\; :\; X\; \backslash to\; Y$ is a linear map and if $q$ is an F-seminorm on $Y,$ then $q\; \backslash circ\; L$ is an F-seminorm on $X.${{sfn|Jarchow|1981|pp=38-42}}
• Let $X\_\backslash Complex$ be a complex vector space and let $X\_\backslash Reals$ denote $X\_\backslash Complex$ considered as a vector space over $\backslash Reals.$ Any F-seminorm on $X\_\backslash Complex$ is also an F-seminorm on $X\_\backslash Reals.${{sfn|Schechter|1996|p=692}}

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

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=91-95}}|math_statement=

Let $p$ be an F-seminorm on a vector space $X.$

Then the map $d\; :\; X\; \backslash times\; X\; \backslash to\; \backslash Reals$ defined by

$d(x,\; y)\; :=\; p(x\; -\; y)$

is a translation invariant pseudometric on $X$ that defines a vector topology $\backslash tau$ on $X.$

If $p$ is an F-norm then $d$ is a metric.

When $X$ is endowed with this topology then $p$ is a continuous map on $X.$

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

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

}}

Suppose that $\backslash mathcal\{L\}$ is a non-empty collection of F-seminorms on a vector space $X$ and for any finite subset $\backslash mathcal\{F\}\; \backslash subseteq\; \backslash mathcal\{L\}$ and any $r\; >\; 0,$ let

The set $\backslash left\backslash \{U\_\{\backslash mathcal\{F\},\; r\}\; ~:~\; r\; >\; 0,\; \backslash mathcal\{F\}\; \backslash subseteq\; \backslash mathcal\{L\},\; \backslash mathcal\{F\}\; \backslash text\{\; finite\; \}\backslash right\backslash \}$ forms a filter base on $X$ that also forms a neighborhood basis at the origin for a vector topology on $X$ denoted by $\backslash tau\_\{\backslash mathcal\{L\}\}.${{sfn|Jarchow|1981|pp=38-42}} Each $U\_\{\backslash mathcal\{F\},\; r\}$ is a balanced and absorbing subset of $X.${{sfn|Jarchow|1981|pp=38-42}} These sets satisfy{{sfn|Jarchow|1981|pp=38-42}}

$$U\_\{\backslash mathcal\{F\},\; r/2\}\; +\; U\_\{\backslash mathcal\{F\},\; r/2\}\; \backslash subseteq\; U\_\{\backslash mathcal\{F\},\; r\}.$$

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

{{anchor|Fréchet combination}}

Suppose that $p\_\{\backslash bull\}\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a family of non-negative subadditive functions on a vector space $X.$

The Fréchet combination{{sfn|Wilansky|2013|pp=15-18}} of $p\_\{\backslash bull\}$ is defined to be the real-valued map

$$p(x)\; :=\; \backslash sum\_\{i=1\}^\{\backslash infty\}\; \backslash frac\{p\_i(x)\}\{2^\{i\}\; \backslash left[\; 1\; +\; p\_i(x)\backslash right]\}.$$

Assume that $p\_\{\backslash bull\}\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is an increasing sequence of seminorms on $X$ and let $p$ be the Fréchet combination of $p\_\{\backslash bull\}.$

Then $p$ is an F-seminorm on $X$ that induces the same locally convex topology as the family $p\_\{\backslash bull\}$ of seminorms.{{sfn|Narici|Beckenstein|2011|p=123}}

Since $p\_\{\backslash bull\}\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is increasing, a basis of open neighborhoods of the origin consists of all sets of the form as $i$ ranges over all positive integers and $r\; >\; 0$ ranges over all positive real numbers.

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

$$d(x,\; y)\; =\; \backslash sum^\{\backslash infty\}\_\{i=1\}\; \backslash frac\{1\}\{2^i\}\; \backslash frac\{p\_i(\; x\; -\; y\; )\}\{1\; +\; p\_i(\; x\; -\; y\; )\}.$$

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}}

If each $p\_i$ is a paranorm then so is $p$ and moreover, $p$ induces the same topology on $X$ as the family $p\_\{\backslash bull\}$ of paranorms.{{sfn|Wilansky|2013|pp=15-18}}

This is also true of the following paranorms on $X$:

• $q(x)\; :=\; \backslash inf\_\{\}\; \backslash left\backslash \{\; \backslash sum\_\{i=1\}^n\; p\_i(x)\; +\; \backslash frac\{1\}\{n\}\; ~:~\; n\; >\; 0\; \backslash text\{\; is\; an\; integer\; \}\backslash right\backslash \}.${{sfn|Wilansky|2013|pp=15-18}}
• $r(x)\; :=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash min\; \backslash left\backslash \{\; \backslash frac\{1\}\{2^n\},\; p\_n(x)\backslash right\backslash \}.${{sfn|Wilansky|2013|pp=15-18}}

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 $R\; :\; [0,\; \backslash infty)\; \backslash to\; [0,\; \backslash infty)$ that has a bounded range, is subadditive (meaning that $R(s\; +\; t)\; \backslash leq\; R(s)\; +\; R(t)$ for all $s,\; t\; \backslash geq\; 0$), and satisfies $R(s)\; =\; 0$ if and only if $s\; =\; 0.$

Examples of bounded remetrization functions include $\backslash arctan\; t,$ $\backslash tanh\; t,$ $t\; \backslash mapsto\; \backslash min\; \backslash \{t,\; 1\backslash \},$ and $t\; \backslash mapsto\; \backslash frac\{t\}\{1\; +\; t\}.${{sfn|Schechter|1996|p=487}}

If $d$ is a pseudometric (respectively, metric) on $X$ and $R$ is a bounded remetrization function then $R\; \backslash circ\; d$ is a bounded pseudometric (respectively, bounded metric) on $X$ that is uniformly equivalent to $d.${{sfn|Schechter|1996|p=487}}

Suppose that $p\_\backslash bull\; =\; \backslash left(p\_i\backslash right)\_\{i=1\}^\backslash infty$ is a family of non-negative F-seminorm on a vector space $X,$ $R$ is a bounded remetrization function, and $r\_\backslash bull\; =\; \backslash left(r\_i\backslash right)\_\{i=1\}^\backslash infty$ is a sequence of positive real numbers whose sum is finite.

Then

$$p(x)\; :=\; \backslash sum\_\{i=1\}^\backslash infty\; r\_i\; R\backslash left(p\_i(x)\backslash right)$$

defines a bounded F-seminorm that is uniformly equivalent to the $p\_\backslash bull.${{sfn|Schechter|1996|pp=692-693}}

It has the property that for any net $x\_\backslash bull\; =\; \backslash left(x\_a\backslash right)\_\{a\; \backslash in\; A\}$ in $X,$ $p\backslash left(x\_\backslash bull\backslash right)\; \backslash to\; 0$ if and only if $p\_i\backslash left(x\_\backslash bull\backslash right)\; \backslash to\; 0$ for all $i.${{sfn|Schechter|1996|pp=692-693}}

$p$ is an F-norm if and only if the $p\_\backslash bull$ separate points on $X.${{sfn|Schechter|1996|pp=692-693}}

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

If $(X,\; \backslash tau)$ is a topological vector space (TVS) (where note in particular that $\backslash tau$ is assumed to be a vector topology) then the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=91-95}}

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

If $(X,\; \backslash tau)$ is a TVS then the following are equivalent:

1. $X$ is metrizable.
2. $X$ is Hausdorff and pseudometrizable.
3. $X$ is Hausdorff and has a countable neighborhood base at the origin.{{sfn|Narici|Beckenstein|2011|pp=91-95}}{{sfn|Jarchow|1981|pp=38-42}}
4. The topology on $X$ is induced by a translation-invariant metric on $X.${{sfn|Narici|Beckenstein|2011|pp=91-95}}
5. The topology on $X$ is induced by an F-norm.{{sfn|Narici|Beckenstein|2011|pp=91-95}}{{sfn|Jarchow|1981|pp=38-42}}
6. The topology on $X$ is induced by a monotone F-norm.{{sfn|Jarchow|1981|pp=38-42}}
7. The topology on $X$ is induced by a total paranorm.

{{Math theorem|name=Birkhoff–Kakutani theorem|math_statement=

If $(X,\; \backslash tau)$ is a topological vector space then the following three conditions are equivalent:{{harvnb|Köthe|1983|loc=section 15.11}}In fact, this is true for topological group, for the proof doesn't use the scalar multiplications.

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

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

}}

If $(X,\; \backslash tau)$ is TVS then the following are equivalent:{{sfn|Narici|Beckenstein|2011|p=123}}

1. $X$ is locally convex and pseudometrizable.
2. $X$ has a countable neighborhood base at the origin consisting of convex sets.
3. The topology of $X$ is induced by a countable family of (continuous) seminorms.
4. The topology of $X$ is induced by a countable increasing sequence of (continuous) seminorms $\backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ (increasing means that for all $i,$ $p\_i\; \backslash geq\; p\_\{i+1\}.$
5. The topology of $X$ is induced by an F-seminorm of the form:

   $$p(x)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; 2^\{-n\}\; \backslash operatorname\{arctan\}\; p\_n(x)$$

   where $\backslash left(p\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ are (continuous) seminorms on $X.${{sfn|Schechter|1996|p=706}}

Let $M$ be a vector subspace of a topological vector space $(X,\; \backslash tau).$

• If $X$ is a pseudometrizable TVS then so is $X\; /\; M.${{sfn|Narici|Beckenstein|2011|pp=91-95}}
• If $X$ is a complete pseudometrizable TVS and $M$ is a closed vector subspace of $X$ then $X\; /\; M$ is complete.{{sfn|Narici|Beckenstein|2011|pp=91-95}}
• If $X$ is metrizable TVS and $M$ is a closed vector subspace of $X$ then $X\; /\; M$ is metrizable.{{sfn|Narici|Beckenstein|2011|pp=91-95}}
• If $p$ is an F-seminorm on $X,$ then the map $P\; :\; X\; /\; M\; \backslash to\; \backslash R$ defined by

  $$P(x\; +\; M)\; :=\; \backslash inf\_\{\}\; \backslash \{\; p(x\; +\; m)\; :\; m\; \backslash in\; M\; \backslash \}$$

  is an F-seminorm on $X\; /\; M$ that induces the usual quotient topology on $X\; /\; M.${{sfn|Narici|Beckenstein|2011|pp=91-95}} If in addition $p$ is an F-norm on $X$ and if $M$ is a closed vector subspace of $X$ then $P$ is an F-norm on $X.${{sfn|Narici|Beckenstein|2011|pp=91-95}}

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

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

The strong dual space $X\_b^\{\backslash prime\}$ of a metrizable locally convex space (such as a Fréchet spaceGabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)) $X$ 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 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 $X$ is a metrizable locally convex space then its strong dual $X\_b^\{\backslash prime\}$ has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.{{sfn|Schaefer|Wolff|1999|p=153}}

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.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

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 {{em|not}} normable must be infinite dimensional.

If $M$ is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then $M$ is normable.{{sfn|Schaefer|Wolff|1999|pp=68-72}}

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

1. $X$ is normable.
2. $X$ has a (von Neumann) bounded neighborhood of the origin.
3. the strong dual space $X^\{\backslash prime\}\_b$ of $X$ is normable.{{sfn|Trèves|2006|p=201}}

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

4. the strong dual space of $X$ is metrizable.{{sfn|Trèves|2006|p=201}}
5. the strong dual space of $X$ is a Fréchet–Urysohn locally convex space.

In particular, if a metrizable locally convex space $X$ (such as a Fréchet space) is {{em|not}} normable then its strong dual space $X^\{\backslash prime\}\_b$ is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space $X^\{\backslash prime\}\_b$ is also neither metrizable nor normable.

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

Suppose that $(X,\; d)$ is a pseudometric space and $B\; \backslash subseteq\; X.$

The set $B$ is metrically bounded or $d$-bounded if there exists a real number $R\; >\; 0$ such that $d(x,\; y)\; \backslash leq\; R$ for all $x,\; y\; \backslash in\; B$;

the smallest such $R$ is then called the diameter or $d$-diameter of $B.${{sfn|Narici|Beckenstein|2011|pp=156-175}}

If $B$ is bounded in a pseudometrizable TVS $X$ then it is metrically bounded;

the converse is in general false but it is true for locally convex metrizable TVSs.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

{{Math theorem|name=Theorem{{sfn|Wilansky|2013|p=57}}|math_statement=

All infinite-dimensional separable complete metrizable TVS are homeomorphic.

}}

• Every metrizable locally convex TVS is a quasibarrelled space,{{sfn|Jarchow|1981|p=222}} bornological space, and a Mackey space.
• 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.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
• If $X$ is a metrizable locally convex space, then the strong dual of $X$ is bornological if and only if it is barreled, if and only if it is infrabarreled.{{sfn|Schaefer|Wolff|1999|p=153}}
• If $X$ is a complete pseudometrizable TVS and $M$ is a closed vector subspace of $X,$ then $X\; /\; M$ is complete.{{sfn|Narici|Beckenstein|2011|pp=91-95}}
• The strong dual of a locally convex metrizable TVS is a webbed space.{{sfn|Narici|Beckenstein|2011|pp=459-483}}
• If $(X,\; \backslash tau)$ and $(X,\; \backslash nu)$ are complete metrizable TVSs (i.e. F-spaces) and if $\backslash nu$ is coarser than $\backslash tau$ then $\backslash tau\; =\; \backslash nu$;{{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 $(X,\; \backslash tau)$ and $(X,\; \backslash nu)$ are both F-spaces but with different topologies, then neither one of $\backslash tau$ and $\backslash nu$ contains the other as a subset. One particular consequence of this is, for example, that if $(X,\; p)$ is a Banach space and $(X,\; q)$ is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of $(X,\; p)$ (i.e. if $p\; \backslash leq\; C\; q$ or if $q\; \backslash leq\; C\; p$ for some constant $C\; >\; 0$), then the only way that $(X,\; q)$ can be a Banach space (i.e. also be complete) is if these two norms $p$ and $q$ are equivalent; if they are not equivalent, then $(X,\; q)$ can not be a Banach space.

  As another consequence, if $(X,\; p)$ is a Banach space and $(X,\; \backslash nu)$ is a Fréchet space, then the map $p\; :\; (X,\; \backslash nu)\; \backslash to\; \backslash R$ is continuous if and only if the Fréchet space $(X,\; \backslash nu)$ {{em|is}} the TVS $(X,\; p)$ (here, the Banach space $(X,\; p)$ is being considered as a TVS, which means that its norm is "forgetten" but its topology is remembered).

• A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.
• Any product of complete metrizable TVSs is a Baire space.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
• A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension $0.${{sfn|Schaefer|Wolff|1999|pp=12-35}}
• 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 {{em|pseudo}}metrizable TVS is a barrelled space and a Baire space (and thus non-meager).{{sfn|Narici|Beckenstein|2011|pp=371-423}}
• The dimension of a complete metrizable TVS is either finite or uncountable.{{sfn|Schaefer|Wolff|1999|pp=12-35}}

{{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 $X$ is a metrizable TVS and $d$ is a metric that defines $X$'s topology, then its possible that $X$ is complete as a TVS (i.e. relative to its uniformity) but the metric $d$ is {{em|not}} a complete metric (such metrics exist even for $X\; =\; \backslash R$).

Thus, if $X$ is a TVS whose topology is induced by a pseudometric $d,$ then the notion of completeness of $X$ (as a TVS) and the notion of completeness of the pseudometric space $(X,\; d)$ are not always equivalent.

The next theorem gives a condition for when they are equivalent:

{{Math theorem|name=Theorem|math_statement=

If $X$ is a pseudometrizable TVS whose topology is induced by a {{em|translation invariant}} pseudometric $d,$ then $d$ is a complete pseudometric on $X$ if and only if $X$ is complete as a TVS.{{sfn|Narici|Beckenstein|2011|pp=47-50}}

}}

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|p=35}}{{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}}|note=Klee|math_statement=

Let $d$ be {{em|any}}Not assumed to be translation-invariant. metric on a vector space $X$ such that the topology $\backslash tau$ induced by $d$ on $X$ makes $(X,\; \backslash tau)$ into a topological vector space. If $(X,\; d)$ is a complete metric space then $(X,\; \backslash tau)$ is a complete-TVS.

}}

{{Math theorem|name=Theorem|math_statement=

If $X$ is a TVS whose topology is induced by a paranorm $p,$ then $X$ is complete if and only if for every sequence $\backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X,$ if then $\backslash sum\_\{i=1\}^\{\backslash infty\}\; x\_i$ converges in $X.${{sfn|Wilansky|2013|pp=56-57}}

}}

If $M$ is a closed vector subspace of a complete pseudometrizable TVS $X,$ then the quotient space $X\; /\; M$ is complete.{{sfn|Narici|Beckenstein|2011|pp=47-66}}

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

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

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

{{Math theorem|name=Banach-Saks theorem{{sfn|Rudin|1991|p=67}}|math_statement=

If $\backslash left(x\_n\backslash right)\_\{n=1\}^\{\backslash infty\}$ is a sequence in a locally convex metrizable TVS $(X,\; \backslash tau)$ that converges {{em|weakly}} to some $x\; \backslash in\; X,$ then there exists a sequence $y\_\{\backslash bull\}\; =\; \backslash left(y\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ in $X$ such that $y\_\{\backslash bull\}\; \backslash to\; x$ in $(X,\; \backslash tau)$ and each $y\_i$ is a convex combination of finitely many $x\_n.$

}}

{{Math theorem|name=Mackey's countability condition{{sfn|Narici|Beckenstein|2011|pp=156-175}}|math_statement=

Suppose that $X$ is a locally convex metrizable TVS and that $\backslash left(B\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ is a countable sequence of bounded subsets of $X.$

Then there exists a bounded subset $B$ of $X$ and a sequence $\backslash left(r\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ of positive real numbers such that $B\_i\; \backslash subseteq\; r\_i\; B$ for all $i.$

}}

Generalized series

As described in this article's section on generalized series, for any $I$-indexed family family $\backslash left(r\_i\backslash right)\_\{i\; \backslash in\; I\}$ of vectors from a TVS $X,$ it is possible to define their sum $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ as the limit of the net of finite partial sums $F\; \backslash in\; \backslash operatorname\{FiniteSubsets\}(I)\; \backslash mapsto\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; F\}\; r\_i$ where the domain $\backslash operatorname\{FiniteSubsets\}(I)$ is directed by $\backslash ,\backslash subseteq.\backslash ,$

If $I\; =\; \backslash N$ and $X\; =\; \backslash Reals,$ for instance, then the generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; \backslash N\}\; r\_i$ converges if and only if $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; r\_i$ converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence).

If a generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ converges in a metrizable TVS, then the set $\backslash left\backslash \{i\; \backslash in\; I\; :\; r\_i\; \backslash neq\; 0\backslash right\backslash \}$ is necessarily countable (that is, either finite or countably infinite);

in other words, all but at most countably many $r\_i$ will be zero and so this generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i\; ~=~\; \backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I\}\{r\_i\; \backslash neq\; 0\}\}\; r\_i$ is actually a sum of at most countably many non-zero terms.

If $X$ is a pseudometrizable TVS and $A$ maps bounded subsets of $X$ to bounded subsets of $Y,$ then $A$ 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 $F\; :\; X\; \backslash to\; Y$ is a linear map between TVSs and $X$ is metrizable then the following are equivalent:

1. $F$ is continuous;
2. $F$ is a (locally) bounded map (that is, $F$ maps (von Neumann) bounded subsets of $X$ to bounded subsets of $Y$);{{sfn|Jarchow|1981|pp=38-42}}
3. $F$ is sequentially continuous;{{sfn|Jarchow|1981|pp=38-42}}
4. the image under $F$ of every null sequence in $X$ 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.
5. $F$ maps null sequences to null sequences;

Open and almost open maps

:Theorem: If $X$ is a complete pseudometrizable TVS, $Y$ is a Hausdorff TVS, and $T\; :\; X\; \backslash to\; Y$ is a closed and almost open linear surjection, then $T$ is an open map.{{sfn|Narici|Beckenstein|2011|pp=466-468}}

:Theorem: If $T\; :\; X\; \backslash to\; Y$ is a surjective linear operator from a locally convex space $X$ onto a barrelled space $Y$ (e.g. every complete pseudometrizable space is barrelled) then $T$ is almost open.{{sfn|Narici|Beckenstein|2011|pp=466-468}}

:Theorem: If $T\; :\; X\; \backslash to\; Y$ is a surjective linear operator from a TVS $X$ onto a Baire space $Y$ then $T$ is almost open.{{sfn|Narici|Beckenstein|2011|pp=466-468}}

:Theorem: Suppose $T\; :\; X\; \backslash to\; Y$ is a continuous linear operator from a complete pseudometrizable TVS $X$ into a Hausdorff TVS $Y.$ If the image of $T$ is non-meager in $Y$ then $T\; :\; X\; \backslash to\; Y$ is a surjective open map and $Y$ is a complete metrizable space.{{sfn|Narici|Beckenstein|2011|pp=466-468}}

{{Main|Hahn-Banach theorem}}

A vector subspace $M$ of a TVS $X$ has the extension property if any continuous linear functional on $M$ can be extended to a continuous linear functional on $X.${{sfn|Narici|Beckenstein|2011|pp=225-273}}

Say that a TVS $X$ has the Hahn-Banach extension property (HBEP) if every vector subspace of $X$ 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 $X$ has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.{{sfn|Narici|Beckenstein|2011|pp=225-273}}

• {{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}}

{{reflist|group=note}}

Proofs

{{reflist|group=proof|refs=

Suppose the net $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \{\backslash textstyle\backslash lim\backslash limits\_\{A\; \backslash in\; \backslash operatorname\{FiniteSubsets\}(I)\}\}\; \backslash \; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; A\}\; r\_i\; =\; \backslash lim\; \backslash left\backslash \{\backslash textstyle\backslash sum\backslash limits\_\{i\backslash in\; A\}\; r\_i\; \backslash ,:\; A\; \backslash subseteq\; I,\; A\; \backslash text\{\; finite\; \}\backslash right\backslash \}$ converges to some point in a metrizable TVS $X,$ where recall that this net's domain is the directed set $(\backslash operatorname\{FiniteSubsets\}(I),\; \backslash subseteq).$

Like every convergent net, this convergent net of partial sums $A\; \backslash mapsto\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; A\}\; r\_i$ is a {{em|Cauchy net}}, which for this particular net means (by definition) that for every neighborhood $W$ of the origin in $X,$ there exists a finite subset $A\_0$ of $I$ such that

$\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; B\}\; r\_i\; -\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; C\}\; r\_i\; \backslash in\; W$ for all finite supersets $B,\; C\; \backslash supseteq\; A\_0;$

this implies that $r\_i\; \backslash in\; W$ for every $i\; \backslash in\; I\; \backslash setminus\; A\_0$ (by taking $B\; :=\; A\_0\; \backslash cup\; \backslash \{i\backslash \}$ and $C\; :=\; A\_0$).

Since $X$ is metrizable, it has a countable neighborhood basis $U\_1,\; U\_2,\; \backslash ldots$ at the origin, whose intersection is necessarily $U\_1\; \backslash cap\; U\_2\; \backslash cap\; \backslash cdots\; =\; \backslash \{0\backslash \}$ (since $X$ is a Hausdorff TVS).

For every positive integer $n\; \backslash in\; \backslash N,$ pick a finite subset $A\_n\; \backslash subseteq\; I$ such that $r\_i\; \backslash in\; U\_n$ for every $i\; \backslash in\; I\; \backslash setminus\; A\_n.$

If $i$ belongs to $(I\; \backslash setminus\; A\_1)\; \backslash cap\; (I\; \backslash setminus\; A\_2)\; \backslash cap\; \backslash cdots\; =\; I\; \backslash setminus\; \backslash left(A\_1\; \backslash cup\; A\_2\; \backslash cup\; \backslash cdots\backslash right)$ then $r\_i$ belongs to $U\_1\; \backslash cap\; U\_2\; \backslash cap\; \backslash cdots\; =\; \backslash \{0\backslash \}.$

Thus $r\_i\; =\; 0$ for every index $i\; \backslash in\; I$ that does not belong to the countable set $A\_1\; \backslash cup\; A\_2\; \backslash cup\; \backslash cdots.$ $\backslash blacksquare$

}}

{{reflist}}

• {{Berberian Lectures in Functional Analysis and Operator Theory}}
• {{Bourbaki Topological Vector Spaces}}
• {{cite journal|last=Bourbaki|first=Nicolas|authorlink=Nicolas Bourbaki|journal=Annales de l'Institut Fourier|language=French|mr=0042609|pages=5–16 (1951)|title=Sur certains espaces vectoriels topologiques|url=|volume=2|year=1950| doi=10.5802/aif.16|doi-access=free}}
• {{Edwards Functional Analysis Theory and Applications}}
• {{Grothendieck Topological Vector Spaces}}
• {{Jarchow Locally Convex Spaces}}
• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Köthe Topological Vector Spaces I}}
• {{Köthe Topological Vector Spaces II}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Rudin Walter Functional Analysis|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{Swartz An Introduction to Functional Analysis}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{Wilansky Modern Methods in Topological Vector Spaces}}
• {{cite book|last=Husain|first=Taqdir|title=Barrelledness in topological and ordered vector spaces|publisher=Springer-Verlag|location=Berlin New York|year=1978|isbn=3-540-09096-7|oclc=4493665 }}

{{Functional analysis}}

{{Topological vector spaces}}

{{Metric spaces}}

Category:Metric spaces

Category:Topological vector spaces