Metrizable topological vector space

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:

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

 

as ${\displaystyle z}$  ranges over ${\displaystyle X}$  and ${\displaystyle r>0}$  ranges over the positive real numbers, forms a basis for a topology on ${\displaystyle X}$  that is called the ${\displaystyle d}$ -topology or the pseudometric topology on ${\displaystyle X}$  induced by ${\displaystyle d.}$ 

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

Theorem^[2] — Suppose that ${\displaystyle X}$  is an additive commutative group. If ${\displaystyle d}$  is a translation invariant pseudometric on ${\displaystyle X}$  then the map ${\displaystyle p(x):=d(x,0)}$  is a value on ${\displaystyle X}$  called the value associated with ${\displaystyle d}$ , and moreover, ${\displaystyle d}$  generates a group topology on ${\displaystyle X}$  (i.e. the ${\displaystyle d}$ -topology on ${\displaystyle X}$  makes ${\displaystyle X}$  into a topological group). Conversely, if ${\displaystyle p}$  is a value on ${\displaystyle X}$  then the map ${\displaystyle d(x,y):=p(x-y)}$  is a translation-invariant pseudometric on ${\displaystyle X}$  and the value associated with ${\displaystyle d}$  is just ${\displaystyle p.}$ 

Pseudometrizable topological groups

Theorem^[2] — If ${\displaystyle (X,\tau )}$  is an additive commutative topological group then the following are equivalent:

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

If ${\displaystyle (X,\tau )}$  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 ${\displaystyle X}$  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 ${\displaystyle U+U\subseteq N.}$ 

Continuity of addition at 0 — If ${\displaystyle (X,+)}$  is a group (as all vector spaces are), ${\displaystyle \tau }$  is a topology on ${\displaystyle X,}$  and ${\displaystyle X\times X}$  is endowed with the product topology, then the addition map ${\displaystyle X\times X\to X}$  (i.e. the map ${\displaystyle (x,y)\mapsto x+y}$ ) is continuous at the origin of ${\displaystyle X\times X}$  if and only if the set of neighborhoods of the origin in ${\displaystyle (X,\tau )}$  is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."^[5]

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

Theorem — Let ${\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }}$  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

Theorem^[11] — Let ${\displaystyle p}$  be an F-seminorm on a vector space ${\displaystyle X.}$  Then the map ${\displaystyle d:X\times X\to \mathbb {R} }$  defined by ${\displaystyle d(x,y):=p(x-y)}$  is a translation invariant pseudometric on ${\displaystyle X}$  that defines a vector topology ${\displaystyle \tau }$  on ${\displaystyle X.}$  If ${\displaystyle p}$  is an F-norm then ${\displaystyle d}$  is a metric. When ${\displaystyle X}$  is endowed with this topology then ${\displaystyle p}$  is a continuous map on ${\displaystyle X.}$ 

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

$${\displaystyle p(x):=\sum _{i=1}^{\mathrm{\infty }}\frac{p_i(x)}{2^i[1}}$$

October 21, 2023