Infunctional analysis and related areas ofmathematics, ametrizable (resp.pseudometrizable)topological vector space (TVS) is a TVS whose topology is induced by a metric (resp.pseudometric). AnLM-space is aninductive limit of a sequence oflocally convex metrizable TVS.

Apseudometric on a set${\displaystyle X}$ is a map${\displaystyle d:X\times X\to \mathbb{R}}$ 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:${\displaystyle d(x,z)\le d(x,y)+d(y,z)\text{ for all }x,y,z\in X.}$

A pseudometric is called ametric 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 aultrapseudometric or astrong pseudometric if it satisfies:

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

Pseudometric space

Apseudometric 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${\displaystyle (X,d)}$ ametric space (resp.ultrapseudometric space) when${\displaystyle d}$ is a metric (resp. ultrapseudometric).

If${\displaystyle d}$ is a pseudometric on a set${\displaystyle X}$ then collection ofopen balls: 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 thepseudometric topology on${\displaystyle X}$ induced by${\displaystyle d.}$

Convention: If${\displaystyle (X,d)}$ is a pseudometric space and${\displaystyle X}$ is treated as atopological 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 calledpseudometrizable (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]

An additivetopological group is an additive group endowed with a topology, called agroup topology, under which addition and negation become continuous operators.

A topology${\displaystyle \tau }$ on a real or complex vector space${\displaystyle X}$ is called avector topology or aTVS topology if it makes the operations of vector addition andscalar multiplication continuous (that is, if it makes${\displaystyle X}$ into atopological vector space).

Everytopological 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, thediscrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

If${\displaystyle X}$ is an additive group then we say that a pseudometric${\displaystyle d}$ on${\displaystyle X}$ istranslation invariant or justinvariant 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. ${\displaystyle d(x,y)=d(x-y,0)\text{ for all }x,y\in X.}$

If${\displaystyle X}$ is atopological group the avalue orG-seminorm on${\displaystyle X}$ (theG stands for Group) is a real-valued map${\displaystyle p:X\to \mathbb{R}}$ with the following properties:^[2]

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

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

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

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

• ${\displaystyle |p(x)-p(y)|\le p(x-y)\text{ for all }x,y\in X.}$^[3]
• ${\displaystyle p(nx)\le np(x)}$ and${\displaystyle \frac{1}{n}p(x)\le p(x/n)}$ 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]

Let${\displaystyle X}$ be a non-trivial (i.e.${\displaystyle X\ne \{0\}}$) real or complex vector space and let${\displaystyle d}$ be the translation-invarianttrivial 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${\displaystyle x\ne y.}$ The topology${\displaystyle \tau }$ that${\displaystyle d}$ induces on${\displaystyle X}$ is thediscrete topology, which makes${\displaystyle (X,\tau )}$ into a commutative topological group under addition but doesnot form a vector topology on${\displaystyle X}$ because${\displaystyle (X,\tau )}$ isdisconnected 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 isnot enough to guarantee a vector topology, which leads us to define paranorms andF-seminorms.

A collection${\displaystyle \mathcal{N}}$ of subsets of a vector space is calledadditive^[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-valuedsubadditive 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 additivetopological groups.

If${\displaystyle X}$ is a vector space over the real or complex numbers then aparanorm on${\displaystyle X}$ is a G-seminorm (defined above)${\displaystyle p:X\to \mathbb{R}}$ on${\displaystyle X}$ that satisfies any of the following additional conditions, each of which begins with "for all sequences${\displaystyle x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}}$ in${\displaystyle X}$ and all convergent sequences of scalars${\displaystyle s_{\bullet }={\left(s_i\right)}_{i=1}^{\mathrm{\infty }}}$":^[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${\displaystyle p\left(s{x}_i\right)\to 0}$ 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${\displaystyle p\left(s{x}_i-sx\right)\to 0}$ 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${\displaystyle p\left(s{x}_i-sx\right)\to 0}$ .

A paranorm is calledtotal if in addition it satisfies:

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

If${\displaystyle p}$ is a paranorm 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 avector 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${\displaystyle p(sx)\le |s|p(x)\text{ for all }x\in X}$ and scalars${\displaystyle s,}$ then${\displaystyle p}$ is absolutely homogeneity (i.e. equality holds)^[8] and thus${\displaystyle p}$ is aseminorm.

• 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).
• Everyseminorm 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${\displaystyle (p\wedge q)(x):=\underset{}{inf}\{p(y)+q(z):x=y+z\text{ with }y,z\in X\}.}$ Moreover,${\displaystyle (p\wedge q)\le p}$ and${\displaystyle (p\wedge q)\le q.}$ This makes the set of paranorms on${\displaystyle X}$ into aconditionally 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${\displaystyle (x,y)\mapsto {\left|x^2-y^2\right|}^{3/2}}$ arenot paranorms on${\displaystyle X:={\mathbb{R}}^2.}$^[8]
• If${\displaystyle x_{\bullet }={\left(x_i\right)}_{i\in I}}$ is aHamel 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 isnot 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]

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

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

AnF-seminorm is called anF-norm if in addition it satisfies:

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

AnF-seminorm is calledmonotone if it satisfies:

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

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

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

Every isometric embedding of oneF-seminormed space into another is atopological embedding, but the converse is not true in general.^[12]

• Every positive scalar multiple of anF-seminorm (resp.F-norm, seminorm) is again anF-seminorm (resp.F-norm, seminorm).
• The sum of finitely manyF-seminorms (resp.F-norms) is anF-seminorm (resp.F-norm).
• If${\displaystyle p}$ and${\displaystyle q}$ areF-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 ofF-seminorms on${\displaystyle X.}$^[12]
• The restriction of anF-seminorm (resp.F-norm) to a vector subspace is anF-seminorm (resp.F-norm).^[9]
• A non-negative real-valued function on${\displaystyle X}$ is a seminorm if and only if it is aconvexF-seminorm, or equivalently, if and only if it is a convex balancedG-seminorm.^[10] In particular, everyseminorm is anF-seminorm.
• For any the map${\displaystyle f}$ on${\displaystyle {\mathbb{R}}^n}$ defined by$${\displaystyle [f\left(x_1,\dots ,x_n\right){]}^p={\left|x_1\right|}^p+\cdots {\left|x_n\right|}^p}$$ is anF-norm that is not a norm.
• If${\displaystyle L:X\to Y}$ is a linear map and if${\displaystyle q}$ is anF-seminorm on${\displaystyle Y,}$ then${\displaystyle q\circ L}$ is anF-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${\displaystyle \mathbb{R}.}$ AnyF-seminorm on${\displaystyle X_{\mathbb{C}}}$ is also anF-seminorm on${\displaystyle X_{\mathbb{R}}.}$^[9]

EveryF-seminorm is a paranorm and every paranorm is equivalent to someF-seminorm.^[7] EveryF-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${\displaystyle x\in X.}$

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

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 abalanced andabsorbing 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 Hausdorffif 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 continuousF-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 ofF-seminorms and${\displaystyle {\tau }_{\mathcal{L}}={\tau }_{\mathcal{F}}.}$^[12]

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

TheFréchet combination^[8] of${\displaystyle p_{\bullet }}$ is defined to be the real-valued map$${\displaystyle p(x):=\sum _{i=1}^{\mathrm{\infty }}\frac{p_i(x)}{2^i\left[1+p_i(x)\right]}.}$$

Assume that${\displaystyle p_{\bullet }={\left(p_i\right)}_{i=1}^{\mathrm{\infty }}}$ 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 anF-seminorm on${\displaystyle X}$ that induces the same locally convex topology as the family${\displaystyle p_{\bullet }}$ of seminorms.^[13]

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

Thetranslation invariantpseudometric on${\displaystyle X}$ induced by thisF-seminorm${\displaystyle p}$ is$${\displaystyle d(x,y)=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{2^i}\frac{p_i(x-y)}{1+p_i(x-y)}.}$$

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

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${\displaystyle X}$:

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

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

Abounded remetrization function^[15] is a continuous non-negative non-decreasing map${\displaystyle R:[0,\mathrm{\infty })\to [0,\mathrm{\infty })}$ that has a bounded range, issubadditive (meaning that${\displaystyle R(s+t)\le R(s)+R(t)}$ for all${\displaystyle s,t\ge 0}$), 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${\displaystyle p_{\bullet }={\left(p_i\right)}_{i=1}^{\mathrm{\infty }}}$ is a family of non-negativeF-seminorm on a vector space${\displaystyle X,}$${\displaystyle R}$ is a bounded remetrization function, and${\displaystyle r_{\bullet }={\left(r_i\right)}_{i=1}^{\mathrm{\infty }}}$ is a sequence of positive real numbers whose sum is finite. Then$${\displaystyle p(x):=\sum _{i=1}^{\mathrm{\infty }}{r}_{i}R\left(p_i(x)\right)}$$defines a boundedF-seminorm that is uniformly equivalent to the${\displaystyle p_{\bullet }.}$^[16] It has the property that for any net${\displaystyle x_{\bullet }={\left(x_a\right)}_{a\in A}}$ in${\displaystyle X,}$${\displaystyle p\left(x_{\bullet }\right)\to 0}$ if and only if${\displaystyle p_i\left(x_{\bullet }\right)\to 0}$ for all${\displaystyle i.}$^[16]${\displaystyle p}$ is anF-norm if and only if the${\displaystyle p_{\bullet }}$ separate points on${\displaystyle X.}$^[16]

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 andabsolutely 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${\displaystyle d.}$

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