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Pseudometrics and metrics

A pseudometric on a set X {\displaystyle X} is a map d : X × X → R {\displaystyle d:X\times X\rightarrow \mathbb {R} } satisfying the following properties:

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

A pseudometric is called a metric if it satisfies:

1. Identity of indiscernibles: for all x , y ∈ X , {\displaystyle x,y\in X,} if d ( x , y ) = 0 {\displaystyle d(x,y)=0} then x = y . {\displaystyle x=y.}

Ultrapseudometric

A pseudometric d {\displaystyle d} on X {\displaystyle X} is called a ultrapseudometric or a strong pseudometric if it satisfies:

1. Strong/Ultrametric triangle inequality: d ( x , z ) ≤ max { d ( x , y ) , d ( y , z ) }  for all  x , y , z ∈ X . {\displaystyle d(x,z)\leq \max\{d(x,y),d(y,z)\}{\text{ for all }}x,y,z\in X.}

Pseudometric space

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

Topology induced by a pseudometric

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

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

Pseudometrizable space

A topological space ( X , τ ) {\displaystyle (X,\tau )} is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) d {\displaystyle d} on X {\displaystyle X} such that τ {\displaystyle \tau } is equal to the topology induced by d . {\displaystyle d.} ¹

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 τ {\displaystyle \tau } on a real or complex vector space X {\displaystyle 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 {\displaystyle X} into a topological vector space).

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

Translation invariant pseudometrics

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

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

Value/G-seminorm

If X {\displaystyle X} is a topological group the a value or G-seminorm on X {\displaystyle X} (the G stands for Group) is a real-valued map p : X → R {\displaystyle p:X\rightarrow \mathbb {R} } with the following properties:²

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

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

1. Total/Positive definite: If p ( x ) = 0 {\displaystyle p(x)=0} then x = 0. {\displaystyle x=0.}

Properties of values

If p {\displaystyle p} is a value on a vector space X {\displaystyle X} then:

• | p ( x ) − p ( y ) | ≤ p ( x − y )  for all  x , y ∈ X . {\displaystyle |p(x)-p(y)|\leq p(x-y){\text{ for all }}x,y\in X.} ³
• p ( n x ) ≤ n p ( x ) {\displaystyle p(nx)\leq np(x)} and 1 n p ( x ) ≤ p ( x / n ) {\displaystyle {\frac {1}{n}}p(x)\leq p(x/n)} for all x ∈ X {\displaystyle x\in X} and positive integers n . {\displaystyle n.} ⁴
• The set { x ∈ X : p ( x ) = 0 } {\displaystyle \{x\in X:p(x)=0\}} is an additive subgroup of X . {\displaystyle X.} ⁵

Equivalence on topological groups

Theorem⁶—Suppose that X {\displaystyle X} is an additive commutative group. If d {\displaystyle d} is a translation invariant pseudometric on X {\displaystyle X} then the map p ( x ) := d ( x , 0 ) {\displaystyle p(x):=d(x,0)} is a value on X {\displaystyle X} called the value associated with d {\displaystyle d} , and moreover, d {\displaystyle d} generates a group topology on X {\displaystyle X} (i.e. the d {\displaystyle d} -topology on X {\displaystyle X} makes X {\displaystyle X} into a topological group). Conversely, if p {\displaystyle p} is a value on X {\displaystyle X} then the map d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)} is a translation-invariant pseudometric on X {\displaystyle X} and the value associated with d {\displaystyle d} is just p . {\displaystyle p.}

Pseudometrizable topological groups

Theorem⁷—If ( X , τ ) {\displaystyle (X,\tau )} is an additive commutative topological group then the following are equivalent:

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

If ( X , τ ) {\displaystyle (X,\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.

An invariant pseudometric that doesn't induce a vector topology

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

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

Additive sequences

A collection N {\displaystyle {\mathcal {N}}} of subsets of a vector space is called additive⁸ if for every N ∈ N , {\displaystyle N\in {\mathcal {N}},} there exists some U ∈ N {\displaystyle U\in {\mathcal {N}}} such that U + U ⊆ N . {\displaystyle U+U\subseteq N.}

Continuity of addition at 0—If ( X , + ) {\displaystyle (X,+)} is a group (as all vector spaces are), τ {\displaystyle \tau } is a topology on X , {\displaystyle X,} and X × X {\displaystyle X\times X} is endowed with the product topology, then the addition map X × X → X {\displaystyle X\times X\to X} (i.e. the map ( x , y ) ↦ x + y {\displaystyle (x,y)\mapsto x+y} ) is continuous at the origin of X × X {\displaystyle X\times X} if and only if the set of neighborhoods of the origin in ( X , τ ) {\displaystyle (X,\tau )} is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."⁹

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

Theorem—Let U ∙ = ( U i ) i = 0 ∞ {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }} be a collection of subsets of a vector space such that 0 ∈ U i {\displaystyle 0\in U_{i}} and U i + 1 + U i + 1 ⊆ U i {\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}} for all i ≥ 0. {\displaystyle i\geq 0.} For all u ∈ U 0 , {\displaystyle u\in U_{0},} let S ( u ) := { n ∙ = ( n 1 , … , n k )   :   k ≥ 1 , n i ≥ 0  for all  i ,  and  u ∈ U n 1 + ⋯ + U n k } . {\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\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\}.}

Define f : X → [ 0 , 1 ] {\displaystyle f:X\to [0,1]} by f ( x ) = 1 {\displaystyle f(x)=1} if x ∉ U 0 {\displaystyle x\not \in U_{0}} and otherwise let f ( x ) := inf { 2 − n 1 + ⋯ 2 − n k   :   n ∙ = ( n 1 , … , n k ) ∈ S ( x ) } . {\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}

Then f {\displaystyle f} is subadditive (meaning f ( x + y ) ≤ f ( x ) + f ( y )  for all  x , y ∈ X {\displaystyle f(x+y)\leq f(x)+f(y){\text{ for all }}x,y\in X} ) and f = 0 {\displaystyle f=0} on ⋂ i ≥ 0 U i , {\displaystyle \bigcap _{i\geq 0}U_{i},} so in particular f ( 0 ) = 0. {\displaystyle f(0)=0.} If all U i {\displaystyle U_{i}} are symmetric sets then f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} and if all U i {\displaystyle U_{i}} are balanced then f ( s x ) ≤ f ( x ) {\displaystyle f(sx)\leq f(x)} for all scalars s {\displaystyle s} such that | s | ≤ 1 {\displaystyle |s|\leq 1} and all x ∈ X . {\displaystyle x\in X.} If X {\displaystyle X} is a topological vector space and if all U i {\displaystyle U_{i}} are neighborhoods of the origin then f {\displaystyle f} is continuous, where if in addition X {\displaystyle X} is Hausdorff and U ∙ {\displaystyle U_{\bullet }} forms a basis of balanced neighborhoods of the origin in X {\displaystyle X} then d ( x , y ) := f ( x − y ) {\displaystyle d(x,y):=f(x-y)} is a metric defining the vector topology on X . {\displaystyle X.}

Proof

Assume that n ∙ = ( n 1 , … , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} always denotes a finite sequence of non-negative integers and use the notation: ∑ 2 − n ∙ := 2 − n 1 + ⋯ + 2 − n k  and  ∑ U n ∙ := U n 1 + ⋯ + U n k . {\displaystyle \sum 2^{-n_{\bullet }}:=2^{-n_{1}}+\cdots +2^{-n_{k}}\quad {\text{ and }}\quad \sum U_{n_{\bullet }}:=U_{n_{1}}+\cdots +U_{n_{k}}.} For any integers n ≥ 0 {\displaystyle n\geq 0} and d > 2 , {\displaystyle d>2,} U n ⊇ U n + 1 + U n + 1 ⊇ U n + 1 + U n + 2 + U n + 2 ⊇ U n + 1 + U n + 2 + ⋯ + U n + d + U n + d + 1 + U n + d + 1 . {\displaystyle 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}.} From this it follows that if n ∙ = ( n 1 , … , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} consists of distinct positive integers then ∑ U n ∙ ⊆ U − 1 + min ( n ∙ ) . {\displaystyle \sum U_{n_{\bullet }}\subseteq U_{-1+\min \left(n_{\bullet }\right)}.} It will now be shown by induction on k {\displaystyle k} that if n ∙ = ( n 1 , … , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} consists of non-negative integers such that ∑ 2 − n ∙ ≤ 2 − M {\displaystyle \sum 2^{-n_{\bullet }}\leq 2^{-M}} for some integer M ≥ 0 {\displaystyle M\geq 0} then ∑ U n ∙ ⊆ U M . {\displaystyle \sum U_{n_{\bullet }}\subseteq U_{M}.} This is clearly true for k = 1 {\displaystyle k=1} and k = 2 {\displaystyle k=2} so assume that k > 2 , {\displaystyle k>2,} which implies that all n i {\displaystyle n_{i}} are positive. If all n i {\displaystyle n_{i}} are distinct then this step is done, and otherwise pick distinct indices i < j {\displaystyle i<j} such that n i = n j {\displaystyle n_{i}=n_{j}} and construct m ∙ = ( m 1 , … , m k − 1 ) {\displaystyle m_{\bullet }=\left(m_{1},\ldots ,m_{k-1}\right)} from n ∙ {\displaystyle n_{\bullet }} by replacing each n i {\displaystyle n_{i}} with n i − 1 {\displaystyle n_{i}-1} and deleting the j th {\displaystyle j^{\text{th}}} element of n ∙ {\displaystyle n_{\bullet }} (all other elements of n ∙ {\displaystyle n_{\bullet }} are transferred to m ∙ {\displaystyle m_{\bullet }} unchanged). Observe that ∑ 2 − n ∙ = ∑ 2 − m ∙ {\displaystyle \sum 2^{-n_{\bullet }}=\sum 2^{-m_{\bullet }}} and ∑ U n ∙ ⊆ ∑ U m ∙ {\displaystyle \sum U_{n_{\bullet }}\subseteq \sum U_{m_{\bullet }}} (because U n i + U n j ⊆ U n i − 1 {\displaystyle U_{n_{i}}+U_{n_{j}}\subseteq U_{n_{i}-1}} ) so by appealing to the inductive hypothesis we conclude that ∑ U n ∙ ⊆ ∑ U m ∙ ⊆ U M , {\displaystyle \sum U_{n_{\bullet }}\subseteq \sum U_{m_{\bullet }}\subseteq U_{M},} as desired. It is clear that f ( 0 ) = 0 {\displaystyle f(0)=0} and that 0 ≤ f ≤ 1 {\displaystyle 0\leq f\leq 1} so to prove that f {\displaystyle f} is subadditive, it suffices to prove that f ( x + y ) ≤ f ( x ) + f ( y ) {\displaystyle f(x+y)\leq f(x)+f(y)} when x , y ∈ X {\displaystyle x,y\in X} are such that f ( x ) + f ( y ) < 1 , {\displaystyle f(x)+f(y)<1,} which implies that x , y ∈ U 0 . {\displaystyle x,y\in U_{0}.} This is an exercise. If all U i {\displaystyle U_{i}} are symmetric then x ∈ ∑ U n ∙ {\displaystyle x\in \sum U_{n_{\bullet }}} if and only if − x ∈ ∑ U n ∙ {\displaystyle -x\in \sum U_{n_{\bullet }}} from which it follows that f ( − x ) ≤ f ( x ) {\displaystyle f(-x)\leq f(x)} and f ( − x ) ≥ f ( x ) . {\displaystyle f(-x)\geq f(x).} If all U i {\displaystyle U_{i}} are balanced then the inequality f ( s x ) ≤ f ( x ) {\displaystyle f(sx)\leq f(x)} for all unit scalars s {\displaystyle s} such that | s | ≤ 1 {\displaystyle |s|\leq 1} is proved similarly. Because f {\displaystyle f} is a nonnegative subadditive function satisfying f ( 0 ) = 0 , {\displaystyle f(0)=0,} as described in the article on sublinear functionals, f {\displaystyle f} is uniformly continuous on X {\displaystyle X} if and only if f {\displaystyle f} is continuous at the origin. If all U i {\displaystyle U_{i}} are neighborhoods of the origin then for any real r > 0 , {\displaystyle r>0,} pick an integer M > 1 {\displaystyle M>1} such that 2 − M < r {\displaystyle 2^{-M}<r} so that x ∈ U M {\displaystyle x\in U_{M}} implies f ( x ) ≤ 2 − M < r . {\displaystyle f(x)\leq 2^{-M}<r.} If the set of all U i {\displaystyle U_{i}} form basis of balanced neighborhoods of the origin then it may be shown that for any n > 1 , {\displaystyle n>1,} there exists some 0 < r ≤ 2 − n {\displaystyle 0<r\leq 2^{-n}} such that f ( x ) < r {\displaystyle f(x)<r} implies x ∈ U n . {\displaystyle x\in U_{n}.} ◼ {\displaystyle \blacksquare }

Paranorms

If X {\displaystyle X} is a vector space over the real or complex numbers then a paranorm on X {\displaystyle X} is a G-seminorm (defined above) p : X → R {\displaystyle p:X\rightarrow \mathbb {R} } on X {\displaystyle X} that satisfies any of the following additional conditions, each of which begins with "for all sequences x ∙ = ( x i ) i = 1 ∞ {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} in X {\displaystyle X} and all convergent sequences of scalars s ∙ = ( s i ) i = 1 ∞ {\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }} ":¹⁰

1. Continuity of multiplication: if s {\displaystyle s} is a scalar and x ∈ X {\displaystyle x\in X} are such that p ( x i − x ) → 0 {\displaystyle p\left(x_{i}-x\right)\to 0} and s ∙ → s , {\displaystyle s_{\bullet }\to s,} then p ( s i x i − s x ) → 0. {\displaystyle p\left(s_{i}x_{i}-sx\right)\to 0.}
2. Both of the conditions:
   • if s ∙ → 0 {\displaystyle s_{\bullet }\to 0} and if x ∈ X {\displaystyle x\in X} is such that p ( x i − x ) → 0 {\displaystyle p\left(x_{i}-x\right)\to 0} then p ( s i x i ) → 0 {\displaystyle p\left(s_{i}x_{i}\right)\to 0} ;
   • if p ( x ∙ ) → 0 {\displaystyle p\left(x_{\bullet }\right)\to 0} then p ( s x i ) → 0 {\displaystyle p\left(sx_{i}\right)\to 0} for every scalar s . {\displaystyle s.}
3. Both of the conditions:
   • if p ( x ∙ ) → 0 {\displaystyle p\left(x_{\bullet }\right)\to 0} and s ∙ → s {\displaystyle s_{\bullet }\to s} for some scalar s {\displaystyle s} then p ( s i x i ) → 0 {\displaystyle p\left(s_{i}x_{i}\right)\to 0} ;
   • if s ∙ → 0 {\displaystyle s_{\bullet }\to 0} then p ( s i x ) → 0  for all  x ∈ X . {\displaystyle p\left(s_{i}x\right)\to 0{\text{ for all }}x\in X.}
4. Separate continuity:¹¹
   • if s ∙ → s {\displaystyle s_{\bullet }\to s} for some scalar s {\displaystyle s} then p ( s x i − s x ) → 0 {\displaystyle p\left(sx_{i}-sx\right)\to 0} for every x ∈ X {\displaystyle x\in X} ;
   • if s {\displaystyle s} is a scalar, x ∈ X , {\displaystyle x\in X,} and p ( x i − x ) → 0 {\displaystyle p\left(x_{i}-x\right)\to 0} then p ( s x i − s x ) → 0 {\displaystyle p\left(sx_{i}-sx\right)\to 0} .

A paranorm is called total if in addition it satisfies:

• Total/Positive definite: p ( x ) = 0 {\displaystyle p(x)=0} implies x = 0. {\displaystyle x=0.}

Properties of paranorms

If p {\displaystyle p} is a paranorm on a vector space X {\displaystyle X} then the map d : X × X → R {\displaystyle d:X\times X\rightarrow \mathbb {R} } defined by d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)} is a translation-invariant pseudometric on X {\displaystyle X} that defines a vector topology on X . {\displaystyle X.} ¹²

If p {\displaystyle p} is a paranorm on a vector space X {\displaystyle X} then:

• the set { x ∈ X : p ( x ) = 0 } {\displaystyle \{x\in X:p(x)=0\}} is a vector subspace of X . {\displaystyle X.} ¹³
• p ( x + n ) = p ( x )  for all  x , n ∈ X {\displaystyle p(x+n)=p(x){\text{ for all }}x,n\in X} with p ( n ) = 0. {\displaystyle p(n)=0.} ¹⁴
• If a paranorm p {\displaystyle p} satisfies p ( s x ) ≤ | s | p ( x )  for all  x ∈ X {\displaystyle p(sx)\leq |s|p(x){\text{ for all }}x\in X} and scalars s , {\displaystyle s,} then p {\displaystyle p} is absolutely homogeneity (i.e. equality holds)¹⁵ and thus p {\displaystyle p} is a seminorm.

Examples of paranorms

• If d {\displaystyle d} is a translation-invariant pseudometric on a vector space X {\displaystyle X} that induces a vector topology τ {\displaystyle \tau } on X {\displaystyle X} (i.e. ( X , τ ) {\displaystyle (X,\tau )} is a TVS) then the map p ( x ) := d ( x − y , 0 ) {\displaystyle p(x):=d(x-y,0)} defines a continuous paranorm on ( X , τ ) {\displaystyle (X,\tau )} ; moreover, the topology that this paranorm p {\displaystyle p} defines in X {\displaystyle X} is τ . {\displaystyle \tau .} ¹⁶
• If p {\displaystyle p} is a paranorm on X {\displaystyle X} then so is the map q ( x ) := p ( x ) / [ 1 + p ( x ) ] . {\displaystyle q(x):=p(x)/[1+p(x)].} ¹⁷
• Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
• Every seminorm is a paranorm.¹⁸
• The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).¹⁹
• The sum of two paranorms is a paranorm.²⁰
• If p {\displaystyle p} and q {\displaystyle q} are paranorms on X {\displaystyle X} then so is ( p ∧ q ) ( x ) := inf { p ( y ) + q ( z ) : x = y + z  with  y , z ∈ X } . {\displaystyle (p\wedge q)(x):=\inf _{}\{p(y)+q(z):x=y+z{\text{ with }}y,z\in X\}.} Moreover, ( p ∧ q ) ≤ p {\displaystyle (p\wedge q)\leq p} and ( p ∧ q ) ≤ q . {\displaystyle (p\wedge q)\leq q.} This makes the set of paranorms on X {\displaystyle X} into a conditionally complete lattice.²¹
• Each of the following real-valued maps are paranorms on X := R 2 {\displaystyle X:=\mathbb {R} ^{2}} :
  • ( x , y ) ↦ | x | {\displaystyle (x,y)\mapsto |x|}
  • ( x , y ) ↦ | x | + | y | {\displaystyle (x,y)\mapsto |x|+|y|}
• The real-valued maps ( x , y ) ↦ | x 2 − y 2 | {\displaystyle (x,y)\mapsto {\sqrt {\left|x^{2}-y^{2}\right|}}} and ( x , y ) ↦ | x 2 − y 2 | 3 / 2 {\displaystyle (x,y)\mapsto \left|x^{2}-y^{2}\right|^{3/2}} are not paranorms on X := R 2 . {\displaystyle X:=\mathbb {R} ^{2}.} ²²
• If x ∙ = ( x i ) i ∈ I {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} is a Hamel basis on a vector space X {\displaystyle X} then the real-valued map that sends x = ∑ i ∈ I s i x i ∈ X {\displaystyle x=\sum _{i\in I}s_{i}x_{i}\in X} (where all but finitely many of the scalars s i {\displaystyle s_{i}} are 0) to ∑ i ∈ I | s i | {\displaystyle \sum _{i\in I}{\sqrt {\left|s_{i}\right|}}} is a paranorm on X , {\displaystyle X,} which satisfies p ( s x ) = | s | p ( x ) {\displaystyle p(sx)={\sqrt {|s|}}p(x)} for all x ∈ X {\displaystyle x\in X} and scalars s . {\displaystyle s.} ²³
• The function p ( x ) := | sin ⁡ ( π x ) | + min { 2 , | x | } {\displaystyle p(x):=|\sin(\pi x)|+\min\{2,|x|\}} is a paranorm on R {\displaystyle \mathbb {R} } that is not balanced but nevertheless equivalent to the usual norm on R . {\displaystyle R.} Note that the function x ↦ | sin ⁡ ( π x ) | {\displaystyle x\mapsto |\sin(\pi x)|} is subadditive.²⁴
• Let X C {\displaystyle X_{\mathbb {C} }} be a complex vector space and let X R {\displaystyle X_{\mathbb {R} }} denote X C {\displaystyle X_{\mathbb {C} }} considered as a vector space over R . {\displaystyle \mathbb {R} .} Any paranorm on X C {\displaystyle X_{\mathbb {C} }} is also a paranorm on X R . {\displaystyle X_{\mathbb {R} }.} ²⁵

F-seminorms

If X {\displaystyle X} is a vector space over the real or complex numbers then an F-seminorm on X {\displaystyle X} (the F {\displaystyle F} stands for Fréchet) is a real-valued map p : X → R {\displaystyle p:X\to \mathbb {R} } with the following four properties: ²⁶

1. Non-negative: p ≥ 0. {\displaystyle p\geq 0.}
2. Subadditive: p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)} for all x , y ∈ X {\displaystyle x,y\in X}
3. Balanced: p ( a x ) ≤ p ( x ) {\displaystyle p(ax)\leq p(x)} for x ∈ X {\displaystyle x\in X} all scalars a {\displaystyle a} satisfying | a | ≤ 1 ; {\displaystyle |a|\leq 1;}
   • This condition guarantees that each set of the form { z ∈ X : p ( z ) ≤ r } {\displaystyle \{z\in X:p(z)\leq r\}} or { z ∈ X : p ( z ) < r } {\displaystyle \{z\in X:p(z)<r\}} for some r ≥ 0 {\displaystyle r\geq 0} is a balanced set.
4. For every x ∈ X , {\displaystyle x\in X,} p ( 1 n x ) → 0 {\displaystyle p\left({\tfrac {1}{n}}x\right)\to 0} as n → ∞ {\displaystyle n\to \infty }
   • The sequence ( 1 n ) n = 1 ∞ {\displaystyle \left({\tfrac {1}{n}}\right)_{n=1}^{\infty }} can be replaced by any positive sequence converging to the zero.²⁷

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

1. Total/Positive definite: p ( x ) = 0 {\displaystyle p(x)=0} implies x = 0. {\displaystyle x=0.}

An F-seminorm is called monotone if it satisfies:

1. Monotone: p ( r x ) < p ( s x ) {\displaystyle p(rx)<p(sx)} for all non-zero x ∈ X {\displaystyle x\in X} and all real s {\displaystyle s} and t {\displaystyle t} such that s < t . {\displaystyle s<t.} ²⁸

F-seminormed spaces

An F-seminormed space (resp. F-normed space)²⁹ is a pair ( X , p ) {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} and an F-seminorm (resp. F-norm) p {\displaystyle p} on X . {\displaystyle X.}

If ( X , p ) {\displaystyle (X,p)} and ( Z , q ) {\displaystyle (Z,q)} are F-seminormed spaces then a map f : X → Z {\displaystyle f:X\to Z} is called an isometric embedding³⁰ if q ( f ( x ) − f ( y ) ) = p ( x , y )  for all  x , y ∈ X . {\displaystyle q(f(x)-f(y))=p(x,y){\text{ for all }}x,y\in X.}

Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.³¹

Examples of F-seminorms

• 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 {\displaystyle p} and q {\displaystyle q} are F-seminorms on X {\displaystyle X} then so is their pointwise supremum x ↦ sup { p ( x ) , q ( x ) } . {\displaystyle x\mapsto \sup\{p(x),q(x)\}.} The same is true of the supremum of any non-empty finite family of F-seminorms on X . {\displaystyle X.} ³²
• The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).³³
• A non-negative real-valued function on X {\displaystyle 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.³⁴ In particular, every seminorm is an F-seminorm.
• For any 0 < p < 1 , {\displaystyle 0<p<1,} the map f {\displaystyle f} on R n {\displaystyle \mathbb {R} ^{n}} defined by [ f ( x 1 , … , x n ) ] p = | x 1 | p + ⋯ | x n | p {\displaystyle [f\left(x_{1},\ldots ,x_{n}\right)]^{p}=\left|x_{1}\right|^{p}+\cdots \left|x_{n}\right|^{p}} is an F-norm that is not a norm.
• If L : X → Y {\displaystyle L:X\to Y} is a linear map and if q {\displaystyle q} is an F-seminorm on Y , {\displaystyle Y,} then q ∘ L {\displaystyle q\circ L} is an F-seminorm on X . {\displaystyle X.} ³⁵
• Let X C {\displaystyle X_{\mathbb {C} }} be a complex vector space and let X R {\displaystyle X_{\mathbb {R} }} denote X C {\displaystyle X_{\mathbb {C} }} considered as a vector space over R . {\displaystyle \mathbb {R} .} Any F-seminorm on X C {\displaystyle X_{\mathbb {C} }} is also an F-seminorm on X R . {\displaystyle X_{\mathbb {R} }.} ³⁶

Properties of F-seminorms

Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.³⁷ Every F-seminorm on a vector space X {\displaystyle X} is a value on X . {\displaystyle X.} In particular, p ( x ) = 0 , {\displaystyle p(x)=0,} and p ( x ) = p ( − x ) {\displaystyle p(x)=p(-x)} for all x ∈ X . {\displaystyle x\in X.}

Topology induced by a single F-seminorm

Theorem³⁸—Let p {\displaystyle p} be an F-seminorm on a vector space X . {\displaystyle X.} Then the map d : X × X → R {\displaystyle d:X\times X\to \mathbb {R} } defined by d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)} is a translation invariant pseudometric on X {\displaystyle X} that defines a vector topology τ {\displaystyle \tau } on X . {\displaystyle X.} If p {\displaystyle p} is an F-norm then d {\displaystyle d} is a metric. When X {\displaystyle X} is endowed with this topology then p {\displaystyle p} is a continuous map on X . {\displaystyle X.}

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

Topology induced by a family of F-seminorms

Suppose that L {\displaystyle {\mathcal {L}}} is a non-empty collection of F-seminorms on a vector space X {\displaystyle X} and for any finite subset F ⊆ L {\displaystyle {\mathcal {F}}\subseteq {\mathcal {L}}} and any r > 0 , {\displaystyle r>0,} let U F , r := ⋂ p ∈ F { x ∈ X : p ( x ) < r } . {\displaystyle U_{{\mathcal {F}},r}:=\bigcap _{p\in {\mathcal {F}}}\{x\in X:p(x)<r\}.}

The set { U F , r   :   r > 0 , F ⊆ L , F  finite  } {\displaystyle \left\{U_{{\mathcal {F}},r}~:~r>0,{\mathcal {F}}\subseteq {\mathcal {L}},{\mathcal {F}}{\text{ finite }}\right\}} forms a filter base on X {\displaystyle X} that also forms a neighborhood basis at the origin for a vector topology on X {\displaystyle X} denoted by τ L . {\displaystyle \tau _{\mathcal {L}}.} ³⁹ Each U F , r {\displaystyle U_{{\mathcal {F}},r}} is a balanced and absorbing subset of X . {\displaystyle X.} ⁴⁰ These sets satisfy⁴¹ U F , r / 2 + U F , r / 2 ⊆ U F , r . {\displaystyle U_{{\mathcal {F}},r/2}+U_{{\mathcal {F}},r/2}\subseteq U_{{\mathcal {F}},r}.}

• τ L {\displaystyle \tau _{\mathcal {L}}} is the coarsest vector topology on X {\displaystyle X} making each p ∈ L {\displaystyle p\in {\mathcal {L}}} continuous.⁴²
• τ L {\displaystyle \tau _{\mathcal {L}}} is Hausdorff if and only if for every non-zero x ∈ X , {\displaystyle x\in X,} there exists some p ∈ L {\displaystyle p\in {\mathcal {L}}} such that p ( x ) > 0. {\displaystyle p(x)>0.} ⁴³
• If F {\displaystyle {\mathcal {F}}} is the set of all continuous F-seminorms on ( X , τ L ) {\displaystyle \left(X,\tau _{\mathcal {L}}\right)} then τ L = τ F . {\displaystyle \tau _{\mathcal {L}}=\tau _{\mathcal {F}}.} ⁴⁴
• If F {\displaystyle {\mathcal {F}}} is the set of all pointwise suprema of non-empty finite subsets of F {\displaystyle {\mathcal {F}}} of L {\displaystyle {\mathcal {L}}} then F {\displaystyle {\mathcal {F}}} is a directed family of F-seminorms and τ L = τ F . {\displaystyle \tau _{\mathcal {L}}=\tau _{\mathcal {F}}.} ⁴⁵

Fréchet combination

Suppose that p ∙ = ( p i ) i = 1 ∞ {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }} is a family of non-negative subadditive functions on a vector space X . {\displaystyle X.}

The Fréchet combination⁴⁶ of p ∙ {\displaystyle p_{\bullet }} is defined to be the real-valued map p ( x ) := ∑ i = 1 ∞ p i ( x ) 2 i [ 1 + p i ( x ) ] . {\displaystyle p(x):=\sum _{i=1}^{\infty }{\frac {p_{i}(x)}{2^{i}\left[1+p_{i}(x)\right]}}.}

As an F-seminorm

Assume that p ∙ = ( p i ) i = 1 ∞ {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }} is an increasing sequence of seminorms on X {\displaystyle X} and let p {\displaystyle p} be the Fréchet combination of p ∙ . {\displaystyle p_{\bullet }.} Then p {\displaystyle p} is an F-seminorm on X {\displaystyle X} that induces the same locally convex topology as the family p ∙ {\displaystyle p_{\bullet }} of seminorms.⁴⁷

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

The translation invariant pseudometric on X {\displaystyle X} induced by this F-seminorm p {\displaystyle p} is d ( x , y ) = ∑ i = 1 ∞ 1 2 i p i ( x − y ) 1 + p i ( x − y ) . {\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\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.⁴⁸

As a paranorm

If each p i {\displaystyle p_{i}} is a paranorm then so is p {\displaystyle p} and moreover, p {\displaystyle p} induces the same topology on X {\displaystyle X} as the family p ∙ {\displaystyle p_{\bullet }} of paranorms.⁴⁹ This is also true of the following paranorms on X {\displaystyle X} :

• q ( x ) := inf { ∑ i = 1 n p i ( x ) + 1 n   :   n > 0  is an integer  } . {\displaystyle q(x):=\inf _{}\left\{\sum _{i=1}^{n}p_{i}(x)+{\frac {1}{n}}~:~n>0{\text{ is an integer }}\right\}.} ⁵⁰
• r ( x ) := ∑ n = 1 ∞ min { 1 2 n , p n ( x ) } . {\displaystyle r(x):=\sum _{n=1}^{\infty }\min \left\{{\frac {1}{2^{n}}},p_{n}(x)\right\}.} ⁵¹

Generalization

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

A bounded remetrization function⁵² is a continuous non-negative non-decreasing map R : [ 0 , ∞ ) → [ 0 , ∞ ) {\displaystyle R:[0,\infty )\to [0,\infty )} that has a bounded range, is subadditive (meaning that R ( s + t ) ≤ R ( s ) + R ( t ) {\displaystyle R(s+t)\leq R(s)+R(t)} for all s , t ≥ 0 {\displaystyle s,t\geq 0} ), and satisfies R ( s ) = 0 {\displaystyle R(s)=0} if and only if s = 0. {\displaystyle s=0.}

Examples of bounded remetrization functions include arctan ⁡ t , {\displaystyle \arctan t,} tanh ⁡ t , {\displaystyle \tanh t,} t ↦ min { t , 1 } , {\displaystyle t\mapsto \min\{t,1\},} and t ↦ t 1 + t . {\displaystyle t\mapsto {\frac {t}{1+t}}.} ⁵³ If d {\displaystyle d} is a pseudometric (respectively, metric) on X {\displaystyle X} and R {\displaystyle R} is a bounded remetrization function then R ∘ d {\displaystyle R\circ d} is a bounded pseudometric (respectively, bounded metric) on X {\displaystyle X} that is uniformly equivalent to d . {\displaystyle d.} ⁵⁴

Suppose that p ∙ = ( p i ) i = 1 ∞ {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }} is a family of non-negative F-seminorm on a vector space X , {\displaystyle X,} R {\displaystyle R} is a bounded remetrization function, and r ∙ = ( r i ) i = 1 ∞ {\displaystyle r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }} is a sequence of positive real numbers whose sum is finite. Then p ( x ) := ∑ i = 1 ∞ r i R ( p i ( x ) ) {\displaystyle p(x):=\sum _{i=1}^{\infty }r_{i}R\left(p_{i}(x)\right)} defines a bounded F-seminorm that is uniformly equivalent to the p ∙ . {\displaystyle p_{\bullet }.} ⁵⁵ It has the property that for any net x ∙ = ( x a ) a ∈ A {\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}} in X , {\displaystyle X,} p ( x ∙ ) → 0 {\displaystyle p\left(x_{\bullet }\right)\to 0} if and only if p i ( x ∙ ) → 0 {\displaystyle p_{i}\left(x_{\bullet }\right)\to 0} for all i . {\displaystyle i.} ⁵⁶ p {\displaystyle p} is an F-norm if and only if the p ∙ {\displaystyle p_{\bullet }} separate points on X . {\displaystyle X.} ⁵⁷

Characterizations

Of (pseudo)metrics induced by (semi)norms

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

Of pseudometrizable TVS

If ( X , τ ) {\displaystyle (X,\tau )} is a topological vector space (TVS) (where note in particular that τ {\displaystyle \tau } is assumed to be a vector topology) then the following are equivalent:⁵⁸

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

Of metrizable TVS

If ( X , τ ) {\displaystyle (X,\tau )} is a TVS then the following are equivalent:

1. X {\displaystyle X} is metrizable.
2. X {\displaystyle X} is Hausdorff and pseudometrizable.
3. X {\displaystyle X} is Hausdorff and has a countable neighborhood base at the origin.⁵⁹⁶⁰
4. The topology on X {\displaystyle X} is induced by a translation-invariant metric on X . {\displaystyle X.} ⁶¹
5. The topology on X {\displaystyle X} is induced by an F-norm.⁶²⁶³
6. The topology on X {\displaystyle X} is induced by a monotone F-norm.⁶⁴
7. The topology on X {\displaystyle X} is induced by a total paranorm.

Birkhoff–Kakutani theorem—If ( X , τ ) {\displaystyle (X,\tau )} is a topological vector space then the following three conditions are equivalent:⁶⁵⁶⁶

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

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

Of locally convex pseudometrizable TVS

If ( X , τ ) {\displaystyle (X,\tau )} is TVS then the following are equivalent:⁶⁷

1. X {\displaystyle X} is locally convex and pseudometrizable.
2. X {\displaystyle X} has a countable neighborhood base at the origin consisting of convex sets.
3. The topology of X {\displaystyle X} is induced by a countable family of (continuous) seminorms.
4. The topology of X {\displaystyle X} is induced by a countable increasing sequence of (continuous) seminorms ( p i ) i = 1 ∞ {\displaystyle \left(p_{i}\right)_{i=1}^{\infty }} (increasing means that for all i , {\displaystyle i,} p i ≥ p i + 1 . {\displaystyle p_{i}\geq p_{i+1}.}
5. The topology of X {\displaystyle X} is induced by an F-seminorm of the form: p ( x ) = ∑ n = 1 ∞ 2 − n arctan ⁡ p n ( x ) {\displaystyle p(x)=\sum _{n=1}^{\infty }2^{-n}\operatorname {arctan} p_{n}(x)} where ( p i ) i = 1 ∞ {\displaystyle \left(p_{i}\right)_{i=1}^{\infty }} are (continuous) seminorms on X . {\displaystyle X.} ⁶⁸

Quotients

Let M {\displaystyle M} be a vector subspace of a topological vector space ( X , τ ) . {\displaystyle (X,\tau ).}

• If X {\displaystyle X} is a pseudometrizable TVS then so is X / M . {\displaystyle X/M.} ⁶⁹
• If X {\displaystyle X} is a complete pseudometrizable TVS and M {\displaystyle M} is a closed vector subspace of X {\displaystyle X} then X / M {\displaystyle X/M} is complete.⁷⁰
• If X {\displaystyle X} is metrizable TVS and M {\displaystyle M} is a closed vector subspace of X {\displaystyle X} then X / M {\displaystyle X/M} is metrizable.⁷¹
• If p {\displaystyle p} is an F-seminorm on X , {\displaystyle X,} then the map P : X / M → R {\displaystyle P:X/M\to \mathbb {R} } defined by P ( x + M ) := inf { p ( x + m ) : m ∈ M } {\displaystyle P(x+M):=\inf _{}\{p(x+m):m\in M\}} is an F-seminorm on X / M {\displaystyle X/M} that induces the usual quotient topology on X / M . {\displaystyle X/M.} ⁷² If in addition p {\displaystyle p} is an F-norm on X {\displaystyle X} and if M {\displaystyle M} is a closed vector subspace of X {\displaystyle X} then P {\displaystyle P} is an F-norm on X . {\displaystyle X.} ⁷³

Examples and sufficient conditions

• Every seminormed space ( X , p ) {\displaystyle (X,p)} is pseudometrizable with a canonical pseudometric given by d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)} for all x , y ∈ X . {\displaystyle x,y\in X.} ⁷⁴.
• If ( X , d ) {\displaystyle (X,d)} is pseudometric TVS with a translation invariant pseudometric d , {\displaystyle d,} then p ( x ) := d ( x , 0 ) {\displaystyle p(x):=d(x,0)} defines a paranorm.⁷⁵ However, if d {\displaystyle d} is a translation invariant pseudometric on the vector space X {\displaystyle X} (without the addition condition that ( X , d ) {\displaystyle (X,d)} is pseudometric TVS), then d {\displaystyle d} need not be either an F-seminorm⁷⁶ nor a paranorm.
• If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.⁷⁷
• If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.⁷⁸
• Suppose X {\displaystyle X} is either a DF-space or an LM-space. If X {\displaystyle X} is a sequential space then it is either metrizable or else a Montel DF-space.

If X {\displaystyle X} is Hausdorff locally convex TVS then X {\displaystyle X} with the strong topology, ( X , b ( X , X ′ ) ) , {\displaystyle \left(X,b\left(X,X^{\prime }\right)\right),} is metrizable if and only if there exists a countable set B {\displaystyle {\mathcal {B}}} of bounded subsets of X {\displaystyle X} such that every bounded subset of X {\displaystyle X} is contained in some element of B . {\displaystyle {\mathcal {B}}.} ⁷⁹

The strong dual space X b ′ {\displaystyle X_{b}^{\prime }} of a metrizable locally convex space (such as a Fréchet space⁸⁰) X {\displaystyle X} is a DF-space.⁸¹ The strong dual of a DF-space is a Fréchet space.⁸² The strong dual of a reflexive Fréchet space is a bornological space.⁸³ The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.⁸⁴ If X {\displaystyle X} is a metrizable locally convex space then its strong dual X b ′ {\displaystyle X_{b}^{\prime }} has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.⁸⁵

Normability

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.⁸⁶ Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is not normable must be infinite dimensional.

If M {\displaystyle M} is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then M {\displaystyle M} is normable.⁸⁷

If X {\displaystyle X} is a Hausdorff locally convex space then the following are equivalent:

1. X {\displaystyle X} is normable.
2. X {\displaystyle X} has a (von Neumann) bounded neighborhood of the origin.
3. the strong dual space X b ′ {\displaystyle X_{b}^{\prime }} of X {\displaystyle X} is normable.⁸⁸

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

1. the strong dual space of X {\displaystyle X} is metrizable.⁸⁹
2. the strong dual space of X {\displaystyle X} is a Fréchet–Urysohn locally convex space.⁹⁰

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

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

Metrically bounded sets and bounded sets

Suppose that ( X , d ) {\displaystyle (X,d)} is a pseudometric space and B ⊆ X . {\displaystyle B\subseteq X.} The set B {\displaystyle B} is metrically bounded or d {\displaystyle d} -bounded if there exists a real number R > 0 {\displaystyle R>0} such that d ( x , y ) ≤ R {\displaystyle d(x,y)\leq R} for all x , y ∈ B {\displaystyle x,y\in B} ; the smallest such R {\displaystyle R} is then called the diameter or d {\displaystyle d} -diameter of B . {\displaystyle B.} ⁹¹ If B {\displaystyle B} is bounded in a pseudometrizable TVS X {\displaystyle X} then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.⁹²

Properties of pseudometrizable TVS

Theorem⁹³—All infinite-dimensional separable complete metrizable TVS are homeomorphic.

• Every metrizable locally convex TVS is a quasibarrelled space,⁹⁴ bornological space, and a Mackey space.
• Every complete pseudometrizable TVS is a barrelled space and a Baire space (and hence non-meager).⁹⁵ However, there exist metrizable Baire spaces that are not complete.⁹⁶
• If X {\displaystyle X} is a metrizable locally convex space, then the strong dual of X {\displaystyle X} is bornological if and only if it is barreled, if and only if it is infrabarreled.⁹⁷
• If X {\displaystyle X} is a complete pseudometrizable TVS and M {\displaystyle M} is a closed vector subspace of X , {\displaystyle X,} then X / M {\displaystyle X/M} is complete.⁹⁸
• The strong dual of a locally convex metrizable TVS is a webbed space.⁹⁹
• If ( X , τ ) {\displaystyle (X,\tau )} and ( X , ν ) {\displaystyle (X,\nu )} are complete metrizable TVSs (i.e. F-spaces) and if ν {\displaystyle \nu } is coarser than τ {\displaystyle \tau } then τ = ν {\displaystyle \tau =\nu } ;¹⁰⁰ this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete.¹⁰¹ Said differently, if ( X , τ ) {\displaystyle (X,\tau )} and ( X , ν ) {\displaystyle (X,\nu )} are both F-spaces but with different topologies, then neither one of τ {\displaystyle \tau } and ν {\displaystyle \nu } contains the other as a subset. One particular consequence of this is, for example, that if ( X , p ) {\displaystyle (X,p)} is a Banach space and ( X , q ) {\displaystyle (X,q)} is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of ( X , p ) {\displaystyle (X,p)} (i.e. if p ≤ C q {\displaystyle p\leq Cq} or if q ≤ C p {\displaystyle q\leq Cp} for some constant C > 0 {\displaystyle C>0} ), then the only way that ( X , q ) {\displaystyle (X,q)} can be a Banach space (i.e. also be complete) is if these two norms p {\displaystyle p} and q {\displaystyle q} are equivalent; if they are not equivalent, then ( X , q ) {\displaystyle (X,q)} can not be a Banach space. As another consequence, if ( X , p ) {\displaystyle (X,p)} is a Banach space and ( X , ν ) {\displaystyle (X,\nu )} is a Fréchet space, then the map p : ( X , ν ) → R {\displaystyle p:(X,\nu )\to \mathbb {R} } is continuous if and only if the Fréchet space ( X , ν ) {\displaystyle (X,\nu )} is the TVS ( X , p ) {\displaystyle (X,p)} (here, the Banach space ( X , p ) {\displaystyle (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.¹⁰²
• Any product of complete metrizable TVSs is a Baire space.¹⁰³
• A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension 0. {\displaystyle 0.} ¹⁰⁴
• A product of pseudometrizable TVSs is pseudometrizable if and only if it all but at most countably many of these TVSs have the trivial topology.
• Every complete pseudometrizable TVS is a barrelled space and a Baire space (and thus non-meager).¹⁰⁵
• The dimension of a complete metrizable TVS is either finite or uncountable.¹⁰⁶

Completeness

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

Theorem—If X {\displaystyle X} is a pseudometrizable TVS whose topology is induced by a translation invariant pseudometric d , {\displaystyle d,} then d {\displaystyle d} is a complete pseudometric on X {\displaystyle X} if and only if X {\displaystyle X} is complete as a TVS.¹⁰⁷

Theorem¹⁰⁸¹⁰⁹ (Klee)—Let d {\displaystyle d} be any¹¹⁰ metric on a vector space X {\displaystyle X} such that the topology τ {\displaystyle \tau } induced by d {\displaystyle d} on X {\displaystyle X} makes ( X , τ ) {\displaystyle (X,\tau )} into a topological vector space. If ( X , d ) {\displaystyle (X,d)} is a complete metric space then ( X , τ ) {\displaystyle (X,\tau )} is a complete-TVS.

Theorem—If X {\displaystyle X} is a TVS whose topology is induced by a paranorm p , {\displaystyle p,} then X {\displaystyle X} is complete if and only if for every sequence ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} in X , {\displaystyle X,} if ∑ i = 1 ∞ p ( x i ) < ∞ {\displaystyle \sum _{i=1}^{\infty }p\left(x_{i}\right)<\infty } then ∑ i = 1 ∞ x i {\displaystyle \sum _{i=1}^{\infty }x_{i}} converges in X . {\displaystyle X.} ¹¹¹

If M {\displaystyle M} is a closed vector subspace of a complete pseudometrizable TVS X , {\displaystyle X,} then the quotient space X / M {\displaystyle X/M} is complete.¹¹² If M {\displaystyle M} is a complete vector subspace of a metrizable TVS X {\displaystyle X} and if the quotient space X / M {\displaystyle X/M} is complete then so is X . {\displaystyle X.} ¹¹³ If X {\displaystyle X} is not complete then M := X , {\displaystyle M:=X,} but not complete, vector subspace of X . {\displaystyle X.}

A Baire separable topological group is metrizable if and only if it is cosmic.¹¹⁴

Subsets and subsequences

• Let M {\displaystyle M} be a separable locally convex metrizable topological vector space and let C {\displaystyle C} be its completion. If S {\displaystyle S} is a bounded subset of C {\displaystyle C} then there exists a bounded subset R {\displaystyle R} of X {\displaystyle X} such that S ⊆ cl C ⁡ R . {\displaystyle S\subseteq \operatorname {cl} _{C}R.} ¹¹⁵
• Every totally bounded subset of a locally convex metrizable TVS X {\displaystyle X} is contained in the closed convex balanced hull of some sequence in X {\displaystyle X} that converges to 0. {\displaystyle 0.}
• In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.¹¹⁶
• If d {\displaystyle d} is a translation invariant metric on a vector space X , {\displaystyle X,} then d ( n x , 0 ) ≤ n d ( x , 0 ) {\displaystyle d(nx,0)\leq nd(x,0)} for all x ∈ X {\displaystyle x\in X} and every positive integer n . {\displaystyle n.} ¹¹⁷
• If ( x i ) i = 1 ∞ {\displaystyle \left(x_{i}\right)_{i=1}^{\infty }} is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence ( r i ) i = 1 ∞ {\displaystyle \left(r_{i}\right)_{i=1}^{\infty }} of positive real numbers diverging to ∞ {\displaystyle \infty } such that ( r i x i ) i = 1 ∞ → 0. {\displaystyle \left(r_{i}x_{i}\right)_{i=1}^{\infty }\to 0.} ¹¹⁸
• A subset of a complete metric space is closed if and only if it is complete. If a space X {\displaystyle X} is not complete, then X {\displaystyle X} is a closed subset of X {\displaystyle X} that is not complete.
• If X {\displaystyle X} is a metrizable locally convex TVS then for every bounded subset B {\displaystyle B} of X , {\displaystyle X,} there exists a bounded disk D {\displaystyle D} in X {\displaystyle X} such that B ⊆ X D , {\displaystyle B\subseteq X_{D},} and both X {\displaystyle X} and the auxiliary normed space X D {\displaystyle X_{D}} induce the same subspace topology on B . {\displaystyle B.} ¹¹⁹

Banach-Saks theorem¹²⁰—If ( x n ) n = 1 ∞ {\displaystyle \left(x_{n}\right)_{n=1}^{\infty }} is a sequence in a locally convex metrizable TVS ( X , τ ) {\displaystyle (X,\tau )} that converges weakly to some x ∈ X , {\displaystyle x\in X,} then there exists a sequence y ∙ = ( y i ) i = 1 ∞ {\displaystyle y_{\bullet }=\left(y_{i}\right)_{i=1}^{\infty }} in X {\displaystyle X} such that y ∙ → x {\displaystyle y_{\bullet }\to x} in ( X , τ ) {\displaystyle (X,\tau )} and each y i {\displaystyle y_{i}} is a convex combination of finitely many x n . {\displaystyle x_{n}.}

Mackey's countability condition¹²¹—Suppose that X {\displaystyle X} is a locally convex metrizable TVS and that ( B i ) i = 1 ∞ {\displaystyle \left(B_{i}\right)_{i=1}^{\infty }} is a countable sequence of bounded subsets of X . {\displaystyle X.} Then there exists a bounded subset B {\displaystyle B} of X {\displaystyle X} and a sequence ( r i ) i = 1 ∞ {\displaystyle \left(r_{i}\right)_{i=1}^{\infty }} of positive real numbers such that B i ⊆ r i B {\displaystyle B_{i}\subseteq r_{i}B} for all i . {\displaystyle i.}

Generalized series

As described in this article's section on generalized series, for any I {\displaystyle I} -indexed family family ( r i ) i ∈ I {\displaystyle \left(r_{i}\right)_{i\in I}} of vectors from a TVS X , {\displaystyle X,} it is possible to define their sum ∑ i ∈ I r i {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} as the limit of the net of finite partial sums F ∈ FiniteSubsets ⁡ ( I ) ↦ ∑ i ∈ F r i {\displaystyle F\in \operatorname {FiniteSubsets} (I)\mapsto \textstyle \sum \limits _{i\in F}r_{i}} where the domain FiniteSubsets ⁡ ( I ) {\displaystyle \operatorname {FiniteSubsets} (I)} is directed by ⊆ . {\displaystyle \,\subseteq .\,} If I = N {\displaystyle I=\mathbb {N} } and X = R , {\displaystyle X=\mathbb {R} ,} for instance, then the generalized series ∑ i ∈ N r i {\displaystyle \textstyle \sum \limits _{i\in \mathbb {N} }r_{i}} converges if and only if ∑ i = 1 ∞ r i {\displaystyle \textstyle \sum \limits _{i=1}^{\infty }r_{i}} converges unconditionally in the usual sense (which for real numbers, is equivalent to absolute convergence). If a generalized series ∑ i ∈ I r i {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}} converges in a metrizable TVS, then the set { i ∈ I : r i ≠ 0 } {\displaystyle \left\{i\in I:r_{i}\neq 0\right\}} is necessarily countable (that is, either finite or countably infinite);¹²² in other words, all but at most countably many r i {\displaystyle r_{i}} will be zero and so this generalized series ∑ i ∈ I r i   =   ∑ r i ≠ 0 i ∈ I r i {\displaystyle \textstyle \sum \limits _{i\in I}r_{i}~=~\textstyle \sum \limits _{\stackrel {i\in I}{r_{i}\neq 0}}r_{i}} is actually a sum of at most countably many non-zero terms.

Linear maps

If X {\displaystyle X} is a pseudometrizable TVS and A {\displaystyle A} maps bounded subsets of X {\displaystyle X} to bounded subsets of Y , {\displaystyle Y,} then A {\displaystyle A} is continuous.¹²³ Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.¹²⁴ Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.¹²⁵

If F : X → Y {\displaystyle F:X\to Y} is a linear map between TVSs and X {\displaystyle X} is metrizable then the following are equivalent:

1. F {\displaystyle F} is continuous;
2. F {\displaystyle F} is a (locally) bounded map (that is, F {\displaystyle F} maps (von Neumann) bounded subsets of X {\displaystyle X} to bounded subsets of Y {\displaystyle Y} );¹²⁶
3. F {\displaystyle F} is sequentially continuous;¹²⁷
4. the image under F {\displaystyle F} of every null sequence in X {\displaystyle X} is a bounded set¹²⁸ where by definition, a null sequence is a sequence that converges to the origin.
5. F {\displaystyle F} maps null sequences to null sequences;

Open and almost open maps

Theorem: If X {\displaystyle X} is a complete pseudometrizable TVS, Y {\displaystyle Y} is a Hausdorff TVS, and T : X → Y {\displaystyle T:X\to Y} is a closed and almost open linear surjection, then T {\displaystyle T} is an open map.¹²⁹ Theorem: If T : X → Y {\displaystyle T:X\to Y} is a surjective linear operator from a locally convex space X {\displaystyle X} onto a barrelled space Y {\displaystyle Y} (e.g. every complete pseudometrizable space is barrelled) then T {\displaystyle T} is almost open.¹³⁰ Theorem: If T : X → Y {\displaystyle T:X\to Y} is a surjective linear operator from a TVS X {\displaystyle X} onto a Baire space Y {\displaystyle Y} then T {\displaystyle T} is almost open.¹³¹ Theorem: Suppose T : X → Y {\displaystyle T:X\to Y} is a continuous linear operator from a complete pseudometrizable TVS X {\displaystyle X} into a Hausdorff TVS Y . {\displaystyle Y.} If the image of T {\displaystyle T} is non-meager in Y {\displaystyle Y} then T : X → Y {\displaystyle T:X\to Y} is a surjective open map and Y {\displaystyle Y} is a complete metrizable space.¹³²

Hahn-Banach extension property

Main article: Hahn-Banach theorem

A vector subspace M {\displaystyle M} of a TVS X {\displaystyle X} has the extension property if any continuous linear functional on M {\displaystyle M} can be extended to a continuous linear functional on X . {\displaystyle X.} ¹³³ Say that a TVS X {\displaystyle X} has the Hahn-Banach extension property (HBEP) if every vector subspace of X {\displaystyle X} has the extension property.¹³⁴

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

Theorem (Kalton)—Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.¹³⁵

If a vector space X {\displaystyle 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.¹³⁶

See also

• Asymmetric norm – Generalization of the concept of a norm
• Complete metric space – Metric geometry
• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• Equivalence of metrics – in mathematics, two metrics on the same underlying set are said to be equivalent if the resulting metric spaces share certain propertiesPages displaying wikidata descriptions as a fallback
• F-space – Topological vector space with a complete translation-invariant metric
• Fréchet space – Locally convex topological vector space that is also a complete metric space
• Generalised metric – Metric geometry
• K-space (functional analysis)
• Locally convex topological vector space – Vector space with a topology defined by convex open sets
• Metric space – Mathematical space with a notion of distance
• Pseudometric space – Generalization of metric spaces in mathematics
• Relation of norms and metrics – Mathematical space with a notion of distancePages displaying short descriptions of redirect targets
• Seminorm – Mathematical function
• Sublinear function – Type of function in linear algebra
• Uniform space – Topological space with a notion of uniform properties
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Notes

Proofs

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56. Schechter 1996, pp. 692–693. - Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. https://search.worldcat.org/oclc/175294365 ↩

57. Schechter 1996, pp. 692–693. - Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. https://search.worldcat.org/oclc/175294365 ↩

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63. Jarchow 1981, pp. 38–42. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. https://search.worldcat.org/oclc/8210342 ↩

64. Jarchow 1981, pp. 38–42. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. https://search.worldcat.org/oclc/8210342 ↩

65. Köthe 1983, section 15.11 - Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. https://mathscinet.ams.org/mathscinet-getitem?mr=0248498 ↩

66. In fact, this is true for topological group, for the proof doesn't use the scalar multiplications. ↩

67. Narici & Beckenstein 2011, p. 123. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

68. Schechter 1996, p. 706. - Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. https://search.worldcat.org/oclc/175294365 ↩

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71. Narici & Beckenstein 2011, pp. 91–95. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

72. Narici & Beckenstein 2011, pp. 91–95. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

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74. Narici & Beckenstein 2011, pp. 115–154. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

75. Wilansky 2013, pp. 15–16. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. https://search.worldcat.org/oclc/849801114 ↩

76. Schaefer & Wolff 1999, pp. 91–92. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

77. Narici & Beckenstein 2011, pp. 156–175. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

78. Narici & Beckenstein 2011, pp. 156–175. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

79. Narici & Beckenstein 2011, pp. 225–273. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

80. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014) https://arxiv.org/pdf/1412.1497.pdf ↩

81. Schaefer & Wolff 1999, p. 154. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

82. Schaefer & Wolff 1999, p. 196. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

83. Schaefer & Wolff 1999, p. 154. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

84. Schaefer & Wolff 1999, p. 153. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

85. Schaefer & Wolff 1999, p. 153. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

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87. Schaefer & Wolff 1999, pp. 68–72. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

88. Trèves 2006, p. 201. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

89. Trèves 2006, p. 201. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

90. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014) https://arxiv.org/pdf/1412.1497.pdf ↩

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93. Wilansky 2013, p. 57. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. https://search.worldcat.org/oclc/849801114 ↩

94. Jarchow 1981, p. 222. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. https://search.worldcat.org/oclc/8210342 ↩

95. Narici & Beckenstein 2011, pp. 371–423. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

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97. Schaefer & Wolff 1999, p. 153. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

98. Narici & Beckenstein 2011, pp. 91–95. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

99. Narici & Beckenstein 2011, pp. 459–483. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

100. Köthe 1969, p. 168. ↩

101. Wilansky 2013, p. 59. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. https://search.worldcat.org/oclc/849801114 ↩

102. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014) https://arxiv.org/pdf/1412.1497.pdf ↩

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104. Schaefer & Wolff 1999, pp. 12–35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

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106. Schaefer & Wolff 1999, pp. 12–35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

107. Narici & Beckenstein 2011, pp. 47–50. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

108. Schaefer & Wolff 1999, p. 35. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

109. Klee, V. L. (1952). "Invariant metrics in groups (solution of a problem of Banach)" (PDF). Proc. Amer. Math. Soc. 3 (3): 484–487. doi:10.1090/s0002-9939-1952-0047250-4. https://www.ams.org/journals/proc/1952-003-03/S0002-9939-1952-0047250-4/S0002-9939-1952-0047250-4.pdf ↩

110. Not assumed to be translation-invariant. ↩

111. Wilansky 2013, pp. 56–57. - Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. https://search.worldcat.org/oclc/849801114 ↩

112. Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

113. Narici & Beckenstein 2011, pp. 47–66. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

114. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014) https://arxiv.org/pdf/1412.1497.pdf ↩

115. Schaefer & Wolff 1999, pp. 190–202. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. https://search.worldcat.org/oclc/840278135 ↩

116. Narici & Beckenstein 2011, pp. 172–173. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

117. Rudin 1991, p. 22. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

118. Rudin 1991, p. 22. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

119. Narici & Beckenstein 2011, pp. 441–457. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

120. Rudin 1991, p. 67. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

121. Narici & Beckenstein 2011, pp. 156–175. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

122. Suppose the net ∑ i ∈ I r i   = def   lim A ∈ FiniteSubsets ⁡ ( I )   ∑ i ∈ A r i = lim { ∑ i ∈ A r i : A ⊆ I , A  finite  } {\textstyle \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\}} converges to some point in a metrizable TVS X , {\displaystyle X,} where recall that this net's domain is the directed set ( FiniteSubsets ⁡ ( I ) , ⊆ ) . {\displaystyle (\operatorname {FiniteSubsets} (I),\subseteq ).} Like every convergent net, this convergent net of partial sums A ↦ ∑ i ∈ A r i {\displaystyle A\mapsto \textstyle \sum \limits _{i\in A}r_{i}} is a Cauchy net, which for this particular net means (by definition) that for every neighborhood W {\displaystyle W} of the origin in X , {\displaystyle X,} there exists a finite subset A 0 {\displaystyle A_{0}} of I {\displaystyle I} such that ∑ i ∈ B r i − ∑ i ∈ C r i ∈ W {\textstyle \textstyle \sum \limits _{i\in B}r_{i}-\textstyle \sum \limits _{i\in C}r_{i}\in W} for all finite supersets B , C ⊇ A 0 ; {\displaystyle B,C\supseteq A_{0};} this implies that r i ∈ W {\displaystyle r_{i}\in W} for every i ∈ I ∖ A 0 {\displaystyle i\in I\setminus A_{0}} (by taking B := A 0 ∪ { i } {\displaystyle B:=A_{0}\cup \{i\}} and C := A 0 {\displaystyle C:=A_{0}} ). Since X {\displaystyle X} is metrizable, it has a countable neighborhood basis U 1 , U 2 , … {\displaystyle U_{1},U_{2},\ldots } at the origin, whose intersection is necessarily U 1 ∩ U 2 ∩ ⋯ = { 0 } {\displaystyle U_{1}\cap U_{2}\cap \cdots =\{0\}} (since X {\displaystyle X} is a Hausdorff TVS). For every positive integer n ∈ N , {\displaystyle n\in \mathbb {N} ,} pick a finite subset A n ⊆ I {\displaystyle A_{n}\subseteq I} such that r i ∈ U n {\displaystyle r_{i}\in U_{n}} for every i ∈ I ∖ A n . {\displaystyle i\in I\setminus A_{n}.} If i {\displaystyle i} belongs to ( I ∖ A 1 ) ∩ ( I ∖ A 2 ) ∩ ⋯ = I ∖ ( A 1 ∪ A 2 ∪ ⋯ ) {\displaystyle (I\setminus A_{1})\cap (I\setminus A_{2})\cap \cdots =I\setminus \left(A_{1}\cup A_{2}\cup \cdots \right)} then r i {\displaystyle r_{i}} belongs to U 1 ∩ U 2 ∩ ⋯ = { 0 } . {\displaystyle U_{1}\cap U_{2}\cap \cdots =\{0\}.} Thus r i = 0 {\displaystyle r_{i}=0} for every index i ∈ I {\displaystyle i\in I} that does not belong to the countable set A 1 ∪ A 2 ∪ ⋯ . {\displaystyle A_{1}\cup A_{2}\cup \cdots .} ◼ {\displaystyle \blacksquare } /wiki/Directed_set ↩

123. Narici & Beckenstein 2011, pp. 156–175. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

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125. Narici & Beckenstein 2011, p. 125. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

126. Jarchow 1981, pp. 38–42. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. https://search.worldcat.org/oclc/8210342 ↩

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128. Jarchow 1981, pp. 38–42. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. https://search.worldcat.org/oclc/8210342 ↩

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134. Narici & Beckenstein 2011, pp. 225–273. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

135. Narici & Beckenstein 2011, pp. 225–273. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

136. Narici & Beckenstein 2011, pp. 225–273. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩