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

Pseudometrics and metrics

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

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

A pseudometric is called a metric if it satisfies:

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

Ultrapseudometric

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

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

Pseudometric space

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

Topology induced by a pseudometric

If ${\displaystyle d}$ is a pseudometric on a set ${\displaystyle X}$ then collection of open balls:

$${\displaystyle B_{r}(z):=\{x\in X:d(x,z)<r\))$$

${\displaystyle z}$

${\displaystyle X}$

${\displaystyle r>0}$

${\displaystyle X}$

${\displaystyle d}$

${\displaystyle X}$

${\displaystyle d.}$

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

Pseudometrizable space

A topological space ${\displaystyle (X,\tau )}$ is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) ${\displaystyle d}$ on ${\displaystyle X}$ such that ${\displaystyle \tau }$ is equal to the topology induced by ${\displaystyle d.}$^[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 ${\displaystyle \tau }$ on a real or complex vector space ${\displaystyle X}$ 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 ${\displaystyle X}$ into a topological vector space).

Every topological vector space (TVS) ${\displaystyle X}$ is an additive commutative topological group but not all group topologies on ${\displaystyle X}$ are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space ${\displaystyle X}$ 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 ${\displaystyle X}$ is an additive group then we say that a pseudometric ${\displaystyle d}$ on ${\displaystyle X}$ is translation invariant or just invariant if it satisfies any of the following equivalent conditions:

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

Value/G-seminorm

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

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

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

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

Properties of values

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

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

Equivalence on topological groups

Pseudometrizable topological groups

An invariant pseudometric that doesn't induce a vector topology

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

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 ${\displaystyle {\mathcal {N))}$ of subsets of a vector space is called additive^[5] if for every ${\displaystyle N\in {\mathcal {N)),}$ there exists some ${\displaystyle U\in {\mathcal {N))}$ such that ${\displaystyle U+U\subseteq N.}$

All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive 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 ${\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty ))$ be a collection of subsets of a vector space such that ${\displaystyle 0\in U_{i))$ and ${\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i))$ for all ${\displaystyle i\geq 0.}$ For all ${\displaystyle u\in U_{0},}$ let

$${\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all ))i,{\text{ and ))u\in U_{n_{1))+\cdots +U_{n_{k))\right\}.}$$

Define ${\displaystyle f:X\to [0,1]}$ by ${\displaystyle f(x)=1}$ if ${\displaystyle x\not \in U_{0))$ and otherwise let

$${\displaystyle f(x):=\inf _{}\left\{2^{-n_{1))+\cdots 2^{-n_{k))~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}$$

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

Proof

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

Paranorms

If ${\displaystyle X}$ is a vector space over the real or complex numbers then a paranorm on ${\displaystyle X}$ is a G-seminorm (defined above) ${\displaystyle p:X\rightarrow \mathbb {R} }$ on ${\displaystyle X}$ that satisfies any of the following additional conditions, each of which begins with "for all sequences ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty ))$ in ${\displaystyle X}$ and all convergent sequences of scalars ${\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty ))$":^[6]

1. Continuity of multiplication: if ${\displaystyle s}$ is a scalar and ${\displaystyle x\in X}$ are such that ${\displaystyle p\left(x_{i}-x\right)\to 0}$ and ${\displaystyle s_{\bullet }\to s,}$ then ${\displaystyle p\left(s_{i}x_{i}-sx\right)\to 0.}$
2. Both of the conditions:
   • if ${\displaystyle s_{\bullet }\to 0}$ and if ${\displaystyle x\in X}$ is such that ${\displaystyle p\left(x_{i}-x\right)\to 0}$ then ${\displaystyle p\left(s_{i}x_{i}\right)\to 0}$;
   • if ${\displaystyle p\left(x_{\bullet }\right)\to 0}$ then ${\displaystyle p\left(sx_{i}\right)\to 0}$ for every scalar ${\displaystyle s.}$
3. Both of the conditions:
   • if ${\displaystyle p\left(x_{\bullet }\right)\to 0}$ and ${\displaystyle s_{\bullet }\to s}$ for some scalar ${\displaystyle s}$ then ${\displaystyle p\left(s_{i}x_{i}\right)\to 0}$;
   • if ${\displaystyle s_{\bullet }\to 0}$ then ${\displaystyle p\left(s_{i}x\right)\to 0{\text{ for all ))x\in X.}$
4. Separate continuity:^[7]
   • if ${\displaystyle s_{\bullet }\to s}$ for some scalar ${\displaystyle s}$ then ${\displaystyle p\left(sx_{i}-sx\right)\to 0}$ for every ${\displaystyle x\in X}$;
   • if ${\displaystyle s}$ is a scalar, ${\displaystyle x\in X,}$ and ${\displaystyle p\left(x_{i}-x\right)\to 0}$ then ${\displaystyle p\left(sx_{i}-sx\right)\to 0}$ .

A paranorm is called total if in addition it satisfies:

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

Properties of paranorms

If ${\displaystyle p}$ is a paranorm on a vector space ${\displaystyle X}$ then the map ${\displaystyle d:X\times X\rightarrow \mathbb {R} }$ defined by ${\displaystyle d(x,y):=p(x-y)}$ is a translation-invariant pseudometric on ${\displaystyle X}$ that defines a vector topology on ${\displaystyle X.}$^[8]

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

• the set ${\displaystyle \{x\in X:p(x)=0\))$ is a vector subspace of ${\displaystyle X.}$^[8]
• ${\displaystyle p(x+n)=p(x){\text{ for all ))x,n\in X}$ with ${\displaystyle p(n)=0.}$^[8]
• If a paranorm ${\displaystyle p}$ satisfies ${\displaystyle p(sx)\leq |s|p(x){\text{ for all ))x\in X}$ and scalars ${\displaystyle s,}$ then ${\displaystyle p}$ is absolutely homogeneity (i.e. equality holds)^[8] and thus ${\displaystyle p}$ is a seminorm.

Examples of paranorms

• If ${\displaystyle d}$ is a translation-invariant pseudometric on a vector space ${\displaystyle X}$ that induces a vector topology ${\displaystyle \tau }$ on ${\displaystyle X}$ (i.e. ${\displaystyle (X,\tau )}$ is a TVS) then the map ${\displaystyle p(x):=d(x-y,0)}$ defines a continuous paranorm on ${\displaystyle (X,\tau )}$; moreover, the topology that this paranorm ${\displaystyle p}$ defines in ${\displaystyle X}$ is ${\displaystyle \tau .}$^[8]
• If ${\displaystyle p}$ is a paranorm on ${\displaystyle X}$ then so is the map ${\displaystyle q(x):=p(x)/[1+p(x)].}$^[8]
• Every positive scalar multiple of a paranorm (resp. total paranorm) is again such a paranorm (resp. total paranorm).
• Every seminorm is a paranorm.^[8]
• The restriction of an paranorm (resp. total paranorm) to a vector subspace is an paranorm (resp. total paranorm).^[9]
• The sum of two paranorms is a paranorm.^[8]
• If ${\displaystyle p}$ and ${\displaystyle q}$ are paranorms on ${\displaystyle X}$ then so is ${\displaystyle (p\wedge q)(x):=\inf _{}\{p(y)+q(z):x=y+z{\text{ with ))y,z\in X\}.}$ Moreover, ${\displaystyle (p\wedge q)\leq p}$ and ${\displaystyle (p\wedge q)\leq q.}$ This makes the set of paranorms on ${\displaystyle X}$ into a conditionally complete lattice.^[8]
• Each of the following real-valued maps are paranorms on ${\displaystyle X:=\mathbb {R} ^{2))$:
  • ${\displaystyle (x,y)\mapsto |x|}$
  • ${\displaystyle (x,y)\mapsto |x|+|y|}$
• The real-valued maps ${\displaystyle (x,y)\mapsto {\sqrt {\left|x^{2}-y^{2}\right|))}$ and ${\displaystyle (x,y)\mapsto \left|x^{2}-y^{2}\right|^{3/2))$ are not a paranorms on ${\displaystyle X:=\mathbb {R} ^{2}.}$^[8]
• If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I))$ is a Hamel basis on a vector space ${\displaystyle X}$ then the real-valued map that sends ${\displaystyle x=\sum _{i\in I}s_{i}x_{i}\in X}$ (where all but finitely many of the scalars ${\displaystyle s_{i))$ are 0) to ${\displaystyle \sum _{i\in I}{\sqrt {\left|s_{i}\right|))}$ is a paranorm on ${\displaystyle X,}$ which satisfies ${\displaystyle p(sx)={\sqrt {|s|))p(x)}$ for all ${\displaystyle x\in X}$ and scalars ${\displaystyle s.}$^[8]
• The function ${\displaystyle p(x):=|\sin(\pi x)|+\min\{2,|x|\))$is a paranorm on ${\displaystyle \mathbb {R} }$ that is not balanced but nevertheless equivalent to the usual norm on ${\displaystyle R.}$ Note that the function ${\displaystyle x\mapsto |\sin(\pi x)|}$ is subadditive.^[10]
• Let ${\displaystyle X_{\mathbb {C} ))$ be a complex vector space and let ${\displaystyle X_{\mathbb {R} ))$ denote ${\displaystyle X_{\mathbb {C} ))$ considered as a vector space over ${\displaystyle \mathbb {R} .}$ Any paranorm on ${\displaystyle X_{\mathbb {C} ))$ is also a paranorm on ${\displaystyle X_{\mathbb {R} }.}$^[9]

F-seminorms

If ${\displaystyle X}$ is a vector space over the real or complex numbers then an F-seminorm on ${\displaystyle X}$ (the ${\displaystyle F}$ stands for Fréchet) is a real-valued map ${\displaystyle p:X\to \mathbb {R} }$ with the following four properties: ^[11]

1. Non-negative: ${\displaystyle p\geq 0.}$
2. Subadditive: ${\displaystyle p(x+y)\leq p(x)+p(y)}$ for all ${\displaystyle x,y\in X}$
3. Balanced: ${\displaystyle p(ax)\leq p(x)}$ for ${\displaystyle x\in X}$ all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|\leq 1;}$
   • This condition guarantees that each set of the form ${\displaystyle \{z\in X:p(z)\leq r\))$ or ${\displaystyle \{z\in X:p(z)<r\))$ for some ${\displaystyle r\geq 0}$ is a balanced set.
4. For every ${\displaystyle x\in X,}$ ${\displaystyle p\left({\tfrac {1}{n))x\right)\to 0}$ as ${\displaystyle n\to \infty }$
   • The sequence ${\displaystyle \left({\tfrac {1}{n))\right)_{n=1}^{\infty ))$ can be replaced by any positive sequence converging to the zero.^[12]

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

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

An F-seminorm is called monotone if it satisfies:

6. Monotone: ${\displaystyle p(rx)<p(sx)}$ for all non-zero ${\displaystyle x\in X}$ and all real ${\displaystyle s}$ and ${\displaystyle t}$ such that ${\displaystyle s<t.}$^[12]

F-seminormed spaces

An F-seminormed space (resp. F-normed space)^[12] is a pair ${\displaystyle (X,p)}$ consisting of a vector space ${\displaystyle X}$ and an F-seminorm (resp. F-norm) ${\displaystyle p}$ on ${\displaystyle X.}$

If ${\displaystyle (X,p)}$ and ${\displaystyle (Z,q)}$ are F-seminormed spaces then a map ${\displaystyle f:X\to Z}$ is called an isometric embedding^[12] if ${\displaystyle q(f(x)-f(y))=p(x,y){\text{ for all ))x,y\in X.}$

Every isometric embedding of one F-seminormed space into another is a topological embedding, but the converse is not true in general.^[12]

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 ${\displaystyle p}$ and ${\displaystyle q}$ are F-seminorms on ${\displaystyle X}$ then so is their pointwise supremum ${\displaystyle x\mapsto \sup\{p(x),q(x)\}.}$ The same is true of the supremum of any non-empty finite family of F-seminorms on ${\displaystyle X.}$^[12]
• The restriction of an F-seminorm (resp. F-norm) to a vector subspace is an F-seminorm (resp. F-norm).^[9]
• A non-negative real-valued function on ${\displaystyle X}$ 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.^[10] In particular, every seminorm is an F-seminorm.
• For any ${\displaystyle 0<p<1,}$ the map ${\displaystyle f}$ on ${\displaystyle \mathbb {R} ^{n))$ defined by
  $${\displaystyle [f\left(x_{1},\ldots ,x_{n}\right)]^{p}=\left|x_{1}\right|^{p}+\cdots \left|x_{n}\right|^{p))$$

  
  is an F-norm that is not a norm.
• If ${\displaystyle L:X\to Y}$ is a linear map and if ${\displaystyle q}$ is an F-seminorm on ${\displaystyle Y,}$ then ${\displaystyle q\circ L}$ is an F-seminorm on ${\displaystyle X.}$^[12]
• Let ${\displaystyle X_{\mathbb {C} ))$ be a complex vector space and let ${\displaystyle X_{\mathbb {R} ))$ denote ${\displaystyle X_{\mathbb {C} ))$ considered as a vector space over ${\displaystyle \mathbb {R} .}$ Any F-seminorm on ${\displaystyle X_{\mathbb {C} ))$ is also an F-seminorm on ${\displaystyle X_{\mathbb {R} }.}$^[9]

Properties of F-seminorms

Every F-seminorm is a paranorm and every paranorm is equivalent to some F-seminorm.^[7] Every F-seminorm on a vector space ${\displaystyle X}$ is a value on ${\displaystyle X.}$ In particular, ${\displaystyle p(x)=0,}$ and ${\displaystyle p(x)=p(-x)}$ for all ${\displaystyle x\in X.}$

Topology induced by a single F-seminorm

Topology induced by a family of F-seminorms

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

$${\displaystyle U_((\mathcal {F)),r}:=\bigcap _{p\in {\mathcal {F))}\{x\in X:p(x)<r\}.}$$

The set ${\displaystyle \left\{U_((\mathcal {F)),r}~:~r>0,{\mathcal {F))\subseteq {\mathcal {L)),{\mathcal {F)){\text{ finite ))\right\))$ forms a filter base on ${\displaystyle X}$ that also forms a neighborhood basis at the origin for a vector topology on ${\displaystyle X}$ denoted by ${\displaystyle \tau _{\mathcal {L)).}$^[12] Each ${\displaystyle U_((\mathcal {F)),r))$ is a balanced and absorbing subset of ${\displaystyle X.}$^[12] These sets satisfy^[12]

$${\displaystyle U_((\mathcal {F)),r/2}+U_((\mathcal {F)),r/2}\subseteq U_((\mathcal {F)),r}.}$$

• ${\displaystyle \tau _{\mathcal {L))}$ is the coarsest vector topology on ${\displaystyle X}$ making each ${\displaystyle p\in {\mathcal {L))}$ continuous.^[12]
• ${\displaystyle \tau _{\mathcal {L))}$ is Hausdorff if and only if for every non-zero ${\displaystyle x\in X,}$ there exists some ${\displaystyle p\in {\mathcal {L))}$ such that ${\displaystyle p(x)>0.}$^[12]
• If ${\displaystyle {\mathcal {F))}$ is the set of all continuous F-seminorms on ${\displaystyle \left(X,\tau _{\mathcal {L))\right)}$ then ${\displaystyle \tau _{\mathcal {L))=\tau _{\mathcal {F)).}$^[12]
• If ${\displaystyle {\mathcal {F))}$ is the set of all pointwise suprema of non-empty finite subsets of ${\displaystyle {\mathcal {F))}$ of ${\displaystyle {\mathcal {L))}$ then ${\displaystyle {\mathcal {F))}$ is a directed family of F-seminorms and ${\displaystyle \tau _{\mathcal {L))=\tau _{\mathcal {F)).}$^[12]

Fréchet combination

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

The Fréchet combination^[8] of ${\displaystyle p_{\bullet ))$ is defined to be the real-valued map

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

As an F-seminorm

Assume that ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty ))$ is an increasing sequence of seminorms on ${\displaystyle X}$ and let ${\displaystyle p}$ be the Fréchet combination of ${\displaystyle p_{\bullet }.}$ Then ${\displaystyle p}$ is an F-seminorm on ${\displaystyle X}$ that induces the same locally convex topology as the family ${\displaystyle p_{\bullet ))$ of seminorms.^[13]

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

The translation invariant pseudometric on ${\displaystyle X}$ induced by this F-seminorm ${\displaystyle p}$ is

$${\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i))}{\frac {p_{i}(x-y)}{1+p_{i}(x-y))).}$$

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

As a paranorm

If each ${\displaystyle p_{i))$ is a paranorm then so is ${\displaystyle p}$ and moreover, ${\displaystyle p}$ induces the same topology on ${\displaystyle X}$ as the family ${\displaystyle p_{\bullet ))$ of paranorms.^[8] This is also true of the following paranorms on ${\displaystyle X}$:

• ${\displaystyle q(x):=\inf _{}\left\{\sum _{i=1}^{n}p_{i}(x)+{\frac {1}{n))~:~n>0{\text{ is an integer ))\right\}.}$^[8]
• ${\displaystyle r(x):=\sum _{n=1}^{\infty }\min \left\((\frac {1}{2^{n))},p_{n}(x)\right\}.}$^[8]

Generalization

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

A bounded remetrization function^[15] is a continuous non-negative non-decreasing map ${\displaystyle R:[0,\infty )\to [0,\infty )}$ that has a bounded range, is subadditive (meaning that ${\displaystyle R(s+t)\leq R(s)+R(t)}$ for all ${\displaystyle s,t\geq 0}$), and satisfies ${\displaystyle R(s)=0}$ if and only if ${\displaystyle s=0.}$

Examples of bounded remetrization functions include ${\displaystyle \arctan t,}$ ${\displaystyle \tanh t,}$ ${\displaystyle t\mapsto \min\{t,1\},}$ and ${\displaystyle t\mapsto {\frac {t}{1+t)).}$^[15] If ${\displaystyle d}$ is a pseudometric (respectively, metric) on ${\displaystyle X}$ and ${\displaystyle R}$ is a bounded remetrization function then ${\displaystyle R\circ d}$ is a bounded pseudometric (respectively, bounded metric) on ${\displaystyle X}$ that is uniformly equivalent to ${\displaystyle d.}$^[15]

Suppose that ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty ))$ is a family of non-negative F-seminorm on a vector space ${\displaystyle X,}$ ${\displaystyle R}$ is a bounded remetrization function, and ${\displaystyle r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty ))$ is a sequence of positive real numbers whose sum is finite. Then

$${\displaystyle p(x):=\sum _{i=1}^{\infty }r_{i}R\left(p_{i}(x)\right)}$$

${\displaystyle p_{\bullet }.}$

${\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A))$

${\displaystyle X,}$

${\displaystyle p\left(x_{\bullet }\right)\to 0}$

${\displaystyle p_{i}\left(x_{\bullet }\right)\to 0}$

${\displaystyle i.}$

${\displaystyle p}$

${\displaystyle p_{\bullet ))$

${\displaystyle X.}$

Characterizations

Of (pseudo)metrics induced by (semi)norms

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

Of pseudometrizable TVS

If ${\displaystyle (X,\tau )}$ is a topological vector space (TVS) (where note in particular that ${\displaystyle \tau }$ is assumed to be a vector topology) then the following are equivalent:^[11]

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

Of metrizable TVS

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

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