Seminorm

In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.

A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.

Definition

Let ${\displaystyle X}$  be a vector space over either the real numbers ${\displaystyle \mathbb {R} }$  or the complex numbers ${\displaystyle \mathbb {C} .}$  A real-valued function ${\displaystyle p:X\to \mathbb {R} }$  is called a seminorm if it satisfies the following two conditions:

1. Subadditivity^[1]/Triangle inequality: ${\displaystyle p(x+y)\leq p(x)+p(y)}$  for all ${\displaystyle x,y\in X.}$ 
2. Absolute homogeneity:^[1] ${\displaystyle p(sx)=|s|p(x)}$  for all ${\displaystyle x\in X}$  and all scalars ${\displaystyle s.}$ 

These two conditions imply that ${\displaystyle p(0)=0}$ ^{[proof 1]} and that every seminorm ${\displaystyle p}$  also has the following property:^{[proof 2]}

3. Nonnegativity:^[1] ${\displaystyle p(x)\geq 0}$  for all ${\displaystyle x\in X.}$ 

Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties.

By definition, a norm on ${\displaystyle X}$  is a seminorm that also separates points, meaning that it has the following additional property:

4. Positive definite/Positive^[1]/Point-separating: whenever ${\displaystyle x\in X}$  satisfies ${\displaystyle p(x)=0,}$  then ${\displaystyle x=0.}$ 

A seminormed space is a pair ${\displaystyle (X,p)}$  consisting of a vector space ${\displaystyle X}$  and a seminorm ${\displaystyle p}$  on ${\displaystyle X.}$  If the seminorm ${\displaystyle p}$  is also a norm then the seminormed space ${\displaystyle (X,p)}$  is called a normed space.

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map ${\displaystyle p:X\to \mathbb {R} }$  is called a sublinear function if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is not necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem. A real-valued function ${\displaystyle p:X\to \mathbb {R} }$  is a seminorm if and only if it is a sublinear and balanced function.

Examples

• The trivial seminorm on ${\displaystyle X,}$  which refers to the constant ${\displaystyle 0}$  map on ${\displaystyle X,}$  induces the indiscrete topology on ${\displaystyle X.}$ 
• Let ${\displaystyle \mu }$  be a measure on a space ${\displaystyle \Omega }$ . For an arbitrary constant ${\displaystyle c\geq 1}$ , let ${\displaystyle X}$  be the set of all functions ${\displaystyle f:\Omega \rightarrow \mathbb {R} }$  for which
  $${\displaystyle \lVert f\rVert _{c}:=\left(\int _{\Omega }|f|^{c}\,d\mu \right)^{1/c}}$$

   
  exists and is finite. It can be shown that ${\displaystyle X}$  is a vector space, and the functional ${\displaystyle \lVert \cdot \rVert _{c}}$  is a seminorm on ${\displaystyle X}$ . However, it is not always a norm (e.g. if ${\displaystyle \Omega =\mathbb {R} }$  and ${\displaystyle \mu }$  is the Lebesgue measure) because ${\displaystyle \lVert h\rVert _{c}=0}$  does not always imply ${\displaystyle h=0}$ . To make ${\displaystyle \lVert \cdot \rVert _{c}}$  a norm, quotient ${\displaystyle X}$  by the closed subspace of functions ${\displaystyle h}$  with ${\displaystyle \lVert h\rVert _{c}=0}$ . The resulting space, ${\displaystyle L^{c}(\mu )}$ , has a norm induced by ${\displaystyle \lVert \cdot \rVert _{c}}$ .
• If ${\displaystyle f}$  is any linear form on a vector space then its absolute value ${\displaystyle |f|,}$  defined by ${\displaystyle x\mapsto |f(x)|,}$  is a seminorm.
• A sublinear function ${\displaystyle f:X\to \mathbb {R} }$  on a real vector space ${\displaystyle X}$  is a seminorm if and only if it is a symmetric function, meaning that ${\displaystyle f(-x)=f(x)}$  for all ${\displaystyle x\in X.}$ 
• Every real-valued sublinear function ${\displaystyle f:X\to \mathbb {R} }$  on a real vector space ${\displaystyle X}$  induces a seminorm ${\displaystyle p:X\to \mathbb {R} }$  defined by ${\displaystyle p(x):=\max\{f(x),f(-x)\}.}$ ^[2]
• Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a vector subspace is once again a seminorm (respectively, norm).
• If ${\displaystyle p:X\to \mathbb {R} }$  and ${\displaystyle q:Y\to \mathbb {R} }$  are seminorms (respectively, norms) on ${\displaystyle X}$  and ${\displaystyle Y}$  then the map ${\displaystyle r:X\times Y\to \mathbb {R} }$  defined by ${\displaystyle r(x,y)=p(x)+q(y)}$  is a seminorm (respectively, a norm) on ${\displaystyle X\times Y.}$  In particular, the maps on ${\displaystyle X\times Y}$  defined by ${\displaystyle (x,y)\mapsto p(x)}$  and ${\displaystyle (x,y)\mapsto q(y)}$  are both seminorms on ${\displaystyle X\times Y.}$ 
• If ${\displaystyle p}$  and ${\displaystyle q}$  are seminorms on ${\displaystyle X}$  then so are^[3]
  $${\displaystyle (p\vee q)(x)=\max\{p(x),q(x)\}}$$

   
  and
  $${\displaystyle (p\wedge q)(x):=\inf\{p(y)+q(z):x=y+z{\text{ with }}y,z\in X\}}$$

   
  where ${\displaystyle p\wedge q\leq p}$  and ${\displaystyle p\wedge q\leq q.}$ ^[4]
• The space of seminorms on ${\displaystyle X}$  is generally not a distributive lattice with respect to the above operations. For example, over ${\displaystyle \mathbb {R} ^{2}}$ , ${\displaystyle p(x,y):=\max(|x|,|y|),q(x,y):=2|x|,r(x,y):=2|y|}$  are such that
  $${\displaystyle ((p\vee q)\wedge (p\vee r))(x,y)=\inf\{\max(2|x_{1}|,|y_{1}|)+\max(|x_{2}|,2|y_{2}|):x=x_{1}+x_{2}{\text{ and }}y=y_{1}+y_{2}\}}$$

   
  while ${\displaystyle (p\vee q\wedge r)(x,y):=\max(|x|,|y|)}$ 
• If ${\displaystyle L:X\to Y}$  is a linear map and ${\displaystyle q:Y\to \mathbb {R} }$  is a seminorm on ${\displaystyle Y,}$  then ${\displaystyle q\circ L:X\to \mathbb {R} }$  is a seminorm on ${\displaystyle X.}$  The seminorm ${\displaystyle q\circ L}$  will be a norm on ${\displaystyle X}$  if and only if ${\displaystyle L}$  is injective and the restriction ${\displaystyle q{\big \vert }_{L(X)}}$  is a norm on ${\displaystyle L(X).}$ 

Minkowski functionals and seminorms

Main article: Minkowski functional

Seminorms on a vector space ${\displaystyle X}$  are intimately tied, via Minkowski functionals, to subsets of ${\displaystyle X}$  that are convex, balanced, and absorbing. Given such a subset ${\displaystyle D}$  of ${\displaystyle X,}$  the Minkowski functional of ${\displaystyle D}$  is a seminorm. Conversely, given a seminorm ${\displaystyle p}$  on ${\displaystyle X,}$  the sets${\displaystyle \{x\in X:p(x)<1\}}$  and ${\displaystyle \{x\in X:p(x)\leq 1\}}$  are convex, balanced, and absorbing and furthermore, the Minkowski functional of these two sets (as well as of any set lying "in between them") is ${\displaystyle p.}$ ^[5]

Algebraic properties

Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity, ${\displaystyle p(0)=0,}$  and for all vectors ${\displaystyle x,y\in X}$ : the reverse triangle inequality: ^[2][6]

$${\displaystyle |p(x)-p(y)|\leq p(x-y)}$$

 

and also ${\textstyle 0\leq \max\{p(x),p(-x)\}}$  and ${\displaystyle p(x)-p(y)\leq p(x-y).}$ ^[2][6]

For any vector ${\displaystyle x\in X}$  and positive real ${\displaystyle r>0:}$ ^[7]

$${\displaystyle x+\{y\in X:p(y)<r\}=\{y\in X:p(x-y)<r\}}$$

 

and furthermore, ${\displaystyle \{x\in X:p(x)<r\}}$  is an absorbing disk in ${\displaystyle X.}$ ^[3]

If ${\displaystyle p}$  is a sublinear function on a real vector space ${\displaystyle X}$  then there exists a linear functional ${\displaystyle f}$  on ${\displaystyle X}$  such that ${\displaystyle f\leq p}$ ^[6] and furthermore, for any linear functional ${\displaystyle g}$  on ${\displaystyle X,}$  ${\displaystyle g\leq p}$  on ${\displaystyle X}$  if and only if ${\displaystyle g^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .}$ ^[6]

Other properties of seminorms

Every seminorm is a balanced function. A seminorm ${\displaystyle p}$  is a norm on ${\displaystyle X}$  if and only if ${\displaystyle \{x\in X:p(x)<1\}}$  does not contain a non-trivial vector subspace.

If ${\displaystyle p:X\to [0,\infty )}$  is a seminorm on ${\displaystyle X}$  then ${\displaystyle \ker p:=p^{-1}(0)}$  is a vector subspace of ${\displaystyle X}$  and for every ${\displaystyle x\in X,}$  ${\displaystyle p}$  is constant on the set ${\displaystyle x+\ker p=\{x+k:p(k)=0\}}$  and equal to ${\displaystyle p(x).}$ ^{[proof 3]}

Furthermore, for any real ${\displaystyle r>0,}$ ^[3]

$${\displaystyle r\{x\in X:p(x)<1\}=\{x\in X:p(x)<r\}=\left\{x\in X:{\tfrac {1}{r}}p(x)<1\right\}.}$$

 

If ${\displaystyle D}$  is a set satisfying ${\displaystyle \{x\in X:p(x)<1\}\subseteq D\subseteq \{x\in X:p(x)\leq 1\}}$  then ${\displaystyle D}$  is absorbing in ${\displaystyle X}$  and ${\displaystyle p=p_{D}}$  where ${\displaystyle p_{D}}$  denotes the Minkowski functional associated with ${\displaystyle D}$  (that is, the gauge of ${\displaystyle D}$ ).^[5] In particular, if ${\displaystyle D}$  is as above and ${\displaystyle q}$  is any seminorm on ${\displaystyle X,}$  then ${\displaystyle q=p}$  if and only if ${\displaystyle \{x\in X:q(x)<1\}\subseteq D\subseteq \{x\in X:q(x)\leq \}.}$ ^[5]

If ${\displaystyle (X,\|\,\cdot \,\|)}$  is a normed space and ${\displaystyle x,y\in X}$  then ${\displaystyle \|x-y\|=\|x-z\|+\|z-y\|}$  for all ${\displaystyle z}$  in the interval ${\displaystyle [x,y].}$ ^[8]

Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.

Relationship to other norm-like concepts

Let ${\displaystyle p:X\to \mathbb {R} }$  be a non-negative function. The following are equivalent:

1. ${\displaystyle p}$  is a seminorm.
2. ${\displaystyle p}$  is a convex ${\displaystyle F}$ -seminorm.
3. ${\displaystyle p}$  is a convex balanced G-seminorm.^[9]

If any of the above conditions hold, then the following are equivalent:

1. ${\displaystyle p}$  is a norm;
2. ${\displaystyle \{x\in X:p(x)<1\}}$  does not contain a non-trivial vector subspace.^[10]
3. There exists a norm on ${\displaystyle X,}$  with respect to which, ${\displaystyle \{x\in X:p(x)<1\}}$  is bounded.

If ${\displaystyle p}$  is a sublinear function on a real vector space ${\displaystyle X}$  then the following are equivalent:^[6]

1. ${\displaystyle p}$  is a linear functional;
2. ${\displaystyle p(x)+p(-x)\leq 0{\text{ for every }}x\in X}$ ;
3. ${\displaystyle p(x)+p(-x)=0{\text{ for every }}x\in X}$ ;

Inequalities involving seminorms

If ${\displaystyle p,q:X\to [0,\infty )}$  are seminorms on ${\displaystyle X}$  then:

• ${\displaystyle p\leq q}$  if and only if ${\displaystyle q(x)\leq 1}$  implies ${\displaystyle p(x)\leq 1.}$ ^[11]
• If ${\displaystyle a>0}$  and ${\displaystyle b>0}$  are such that ${\displaystyle p(x)<a}$  implies ${\displaystyle q(x)\leq b,}$  then ${\displaystyle aq(x)\leq bp(x)}$  for all ${\displaystyle x\in X.}$  ^[12]
• Suppose ${\displaystyle a}$  and ${\displaystyle b}$  are positive real numbers and ${\displaystyle q,p_{1},\ldots ,p_{n}}$  are seminorms on ${\displaystyle X}$  such that for every ${\displaystyle x\in X,}$  if ${\displaystyle \max\{p_{1}(x),\ldots ,p_{n}(x)\}<a}$  then ${\displaystyle q(x)<b.}$  Then ${\displaystyle aq\leq b\left(p_{1}+\cdots +p_{n}\right).}$ ^[10]
• If ${\displaystyle X}$  is a vector space over the reals and ${\displaystyle f}$  is a non-zero linear functional on ${\displaystyle X,}$  then ${\displaystyle f\leq p}$  if and only if ${\displaystyle \varnothing =f^{-1}(1)\cap \{x\in X:p(x)<1\}.}$ ^[11]

If ${\displaystyle p}$  is a seminorm on ${\displaystyle X}$  and ${\displaystyle f}$  is a linear functional on ${\displaystyle X}$  then:

• ${\displaystyle |f|\leq p}$  on ${\displaystyle X}$  if and only if ${\displaystyle \operatorname {Re} f\leq p}$  on ${\displaystyle X}$  (see footnote for proof).^[13][14]
• ${\displaystyle f\leq p}$  on ${\displaystyle X}$  if and only if ${\displaystyle f^{-1}(1)\cap \{x\in X:p(x)<1=\varnothing \}.}$ ^[6][11]
• If ${\displaystyle a>0}$  and ${\displaystyle b>0}$  are such that ${\displaystyle p(x)<a}$  implies ${\displaystyle f(x)\neq b,}$  then ${\displaystyle a|f(x)|\leq bp(x)}$  for all ${\displaystyle x\in X.}$ ^[12]

Hahn–Banach theorem for seminorms

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:

If ${\displaystyle M}$  is a vector subspace of a seminormed space ${\displaystyle (X,p)}$  and if ${\displaystyle f}$  is a continuous linear functional on ${\displaystyle M,}$  then ${\displaystyle f}$  may be extended to a continuous linear functional ${\displaystyle F}$  on ${\displaystyle X}$  that has the same norm as ${\displaystyle f.}$ ^[15]

A similar extension property also holds for seminorms:

Theorem^[16][12] (Extending seminorms) — If ${\displaystyle M}$  is a vector subspace of ${\displaystyle X,}$  ${\displaystyle p}$  is a seminorm on ${\displaystyle M,}$  and ${\displaystyle q}$  is a seminorm on ${\displaystyle X}$  such that ${\displaystyle p\leq q{\big \vert }_{M},}$  then there exists a seminorm ${\displaystyle P}$  on ${\displaystyle X}$  such that ${\displaystyle P{\big \vert }_{M}=p}$  and ${\displaystyle P\leq q.}$ 

Proof: Let ${\displaystyle S}$  be the convex hull of ${\displaystyle \{m\in M:p(m)\leq 1\}\cup \{x\in X:q(x)\leq 1\}.}$  Then ${\displaystyle S}$  is an absorbing disk in ${\displaystyle X}$  and so the Minkowski functional ${\displaystyle P}$  of ${\displaystyle S}$  is a seminorm on ${\displaystyle X.}$  This seminorm satisfies ${\displaystyle p=P}$  on ${\displaystyle M}$  and ${\displaystyle P\leq q}$  on ${\displaystyle X.}$  ${\displaystyle \blacksquare }$ 

Topologies of seminormed spaces

Pseudometrics and the induced topology

A seminorm ${\displaystyle p}$  on ${\displaystyle X}$  induces a topology, called the seminorm-induced topology, via the canonical translation-invariant pseudometric ${\displaystyle d_{p}:X\times X\to \mathbb {R} }$ ; ${\displaystyle d_{p}(x,y):=p(x-y)=p(y-x).}$  This topology is Hausdorff if and only if ${\displaystyle d_{p}}$  is a metric, which occurs if and only if ${\displaystyle p}$  is a norm.^[4] This topology makes ${\displaystyle X}$  into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis at the origin consisting of the following open balls (or the closed balls) centered at the origin:

$${\displaystyle \{x\in X:p(x)<r\}\quad {\text{ or }}\quad \{x\in X:p(x)\leq r\}}$$

 

as ${\displaystyle r>0}$  ranges over the positive reals. Every seminormed space ${\displaystyle (X,p)}$  should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called seminormable.

Equivalently, every vector space ${\displaystyle X}$  with seminorm ${\displaystyle p}$  induces a vector space quotient ${\displaystyle X/W,}$  where ${\displaystyle W}$  is the subspace of ${\displaystyle X}$  consisting of all vectors ${\displaystyle x\in X}$  with ${\displaystyle p(x)=0.}$  Then ${\displaystyle X/W}$  carries a norm defined by ${\displaystyle p(x+W)=p(v).}$  The resulting topology, pulled back to ${\displaystyle X,}$  is precisely the topology induced by ${\displaystyle p.}$ 

Any seminorm-induced topology makes ${\displaystyle X}$  locally convex, as follows. If ${\displaystyle p}$  is a seminorm on ${\displaystyle X}$  and ${\displaystyle r\in \mathbb {R} ,}$  call the set ${\displaystyle \{x\in X:p(x)<r\}}$  the open ball of radius ${\displaystyle r}$  about the origin; likewise the closed ball of radius ${\displaystyle r}$  is ${\displaystyle \{x\in X:p(x)\leq r\}.}$  The set of all open (resp. closed) ${\displaystyle p}$ -balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the ${\displaystyle p}$ -topology on ${\displaystyle X.}$ 

Stronger, weaker, and equivalent seminorms

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If ${\displaystyle p}$  and ${\displaystyle q}$  are seminorms on ${\displaystyle X,}$  then we say that ${\displaystyle q}$  is stronger than ${\displaystyle p}$  and that ${\displaystyle p}$  is weaker than ${\displaystyle q}$  if any of the following equivalent conditions holds:

1. The topology on ${\displaystyle X}$  induced by ${\displaystyle q}$  is finer than the topology induced by ${\displaystyle p.}$ 
2. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$  is a sequence in ${\displaystyle X,}$  then ${\displaystyle q\left(x_{\bullet }\right):=\left(q\left(x_{i}\right)\right)_{i=1}^{\infty }\to 0}$  in ${\displaystyle \mathbb {R} }$  implies ${\displaystyle p\left(x_{\bullet }\right)\to 0}$  in ${\displaystyle \mathbb {R} .}$ ^[4]
3. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$  is a net in ${\displaystyle X,}$  then ${\displaystyle q\left(x_{\bullet }\right):=\left(q\left(x_{i}\right)\right)_{i\in I}\to 0}$  in ${\displaystyle \mathbb {R} }$  implies ${\displaystyle p\left(x_{\bullet }\right)\to 0}$  in ${\displaystyle \mathbb {R} .}$ 
4. ${\displaystyle p}$  is bounded on ${\displaystyle \{x\in X:q(x)<1\}.}$ ^[4]
5. If ${\displaystyle \inf {}\{q(x):p(x)=1,x\in X\}=0}$  then ${\displaystyle p(x)=0}$  for all ${\displaystyle x\in X.}$ ^[4]
6. There exists a real ${\displaystyle K>0}$  such that ${\displaystyle p\leq Kq}$  on ${\displaystyle X.}$ ^[4]

The seminorms ${\displaystyle p}$  and ${\displaystyle q}$  are called equivalent if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions:

1. The topology on ${\displaystyle X}$  induced by ${\displaystyle q}$  is the same as the topology induced by ${\displaystyle p.}$ 
2. ${\displaystyle q}$  is stronger than ${\displaystyle p}$  and ${\displaystyle p}$  is stronger than ${\displaystyle q.}$ ^[4]
3. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$  is a sequence in ${\displaystyle X}$  then ${\displaystyle q\left(x_{\bullet }\right):=\left(q\left(x_{i}\right)\right)_{i=1}^{\infty }\to 0}$  if and only if ${\displaystyle p\left(x_{\bullet }\right)\to 0.}$ 
4. There exist positive real numbers ${\displaystyle r>0}$  and ${\displaystyle R>0}$  such that ${\displaystyle rq\leq p\leq Rq.}$ 

Normability and seminormability

See also: Normed space and Local boundedness § locally bounded topological vector space

A topological vector space (TVS) is said to be a seminormable space (respectively, a normable space) if its topology is induced by a single seminorm (resp. a single norm). A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T₁ (because a TVS is Hausdorff if and only if it is a T₁ space). A locally bounded topological vector space is a topological vector space that possesses a bounded neighborhood of the origin.

Normability of topological vector spaces is characterized by Kolmogorov's normability criterion. A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.^[17] Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.^[18] A TVS is normable if and only if it is a T₁ space and admits a bounded convex neighborhood of the origin.

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

1. ${\displaystyle X}$  is normable.
2. ${\displaystyle X}$  is seminormable.
3. ${\displaystyle X}$  has a bounded neighborhood of the origin.
4. The strong dual ${\displaystyle X_{b}^{\prime }}$  of ${\displaystyle X}$  is normable.^[19]
5. The strong dual ${\displaystyle X_{b}^{\prime }}$  of ${\displaystyle X}$  is metrizable.^[19]

Furthermore, ${\displaystyle X}$  is finite dimensional if and only if ${\displaystyle X_{\sigma }^{\prime }}$  is normable (here ${\displaystyle X_{\sigma }^{\prime }}$  denotes ${\displaystyle X^{\prime }}$  endowed with the weak-* topology).

The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional).^[18]

Topological properties

• If ${\displaystyle X}$  is a TVS and ${\displaystyle p}$  is a continuous seminorm on ${\displaystyle X,}$  then the closure of ${\displaystyle \{x\in X:p(x)<r\}}$  in ${\displaystyle X}$  is equal to ${\displaystyle \{x\in X:p(x)\leq r\}.}$ ^[3]
• The closure of ${\displaystyle \{0\}}$  in a locally convex space ${\displaystyle X}$  whose topology is defined by a family of continuous seminorms ${\displaystyle {\mathcal {P}}}$  is equal to ${\displaystyle \bigcap _{p\in {\mathcal {P}}}p^{-1}(0).}$ ^[11]
• A subset ${\displaystyle S}$  in a seminormed space ${\displaystyle (X,p)}$  is bounded if and only if ${\displaystyle p(S)}$  is bounded.^[20]
• If ${\displaystyle (X,p)}$  is a seminormed space then the locally convex topology that ${\displaystyle p}$  induces on ${\displaystyle X}$  makes ${\displaystyle X}$  into a pseudometrizable TVS with a canonical pseudometric given by ${\displaystyle d(x,y):=p(x-y)}$  for all ${\displaystyle x,y\in X.}$ ^[21]
• The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces are trivial (that is, 0-dimensional).^[18]

Continuity of seminorms

If ${\displaystyle p}$  is a seminorm on a topological vector space ${\displaystyle X,}$  then the following are equivalent:^[5]

1. ${\displaystyle p}$  is continuous.
2. ${\displaystyle p}$  is continuous at 0;^[3]
3. ${\displaystyle \{x\in X:p(x)<1\}}$  is open in ${\displaystyle X}$ ;^[3]
4. ${\displaystyle \{x\in X:p(x)\leq 1\}}$  is closed neighborhood of 0 in ${\displaystyle X}$ ;^[3]
5. ${\displaystyle p}$  is uniformly continuous on ${\displaystyle X}$ ;^[3]
6. There exists a continuous seminorm ${\displaystyle q}$  on ${\displaystyle X}$  such that ${\displaystyle p\leq q.}$ ^[3]

In particular, if ${\displaystyle (X,p)}$  is a seminormed space then a seminorm ${\displaystyle q}$  on ${\displaystyle X}$  is continuous if and only if ${\displaystyle q}$  is dominated by a positive scalar multiple of ${\displaystyle p.}$ ^[3]

If ${\displaystyle X}$  is a real TVS, ${\displaystyle f}$  is a linear functional on ${\displaystyle X,}$  and ${\displaystyle p}$  is a continuous seminorm (or more generally, a sublinear function) on ${\displaystyle X,}$  then ${\displaystyle f\leq p}$  on ${\displaystyle X}$  implies that ${\displaystyle f}$  is continuous.^[6]

Continuity of linear maps

If ${\displaystyle F:(X,p)\to (Y,q)}$  is a map between seminormed spaces then let^[15]

$${\displaystyle \|F\|_{p,q}:=\sup\{q(F(x)):p(x)\leq 1,x\in X\}.}$$

 

If ${\displaystyle F:(X,p)\to (Y,q)}$  is a linear map between seminormed spaces then the following are equivalent:

1. ${\displaystyle F}$  is continuous;
2. ${\displaystyle \|F\|_{p,q}<\infty }$ ;^[15]
3. There exists a real ${\displaystyle K\geq 0}$  such that ${\displaystyle p\leq Kq}$ ;^[15]
   • In this case, ${\displaystyle \|F\|_{p,q}\leq K.}$ 

If ${\displaystyle F}$  is continuous then ${\displaystyle q(F(x))\leq \|F\|_{p,q}p(x)}$  for all ${\displaystyle x\in X.}$ ^[15]

The space of all continuous linear maps ${\displaystyle F:(X,p)\to (Y,q)}$  between seminormed spaces is itself a seminormed space under the seminorm ${\displaystyle \|F\|_{p,q}.}$  This seminorm is a norm if ${\displaystyle q}$  is a norm.^[15]

Generalizations

The concept of norm in composition algebras does not share the usual properties of a norm.

A composition algebra ${\displaystyle (A,*,N)}$  consists of an algebra over a field ${\displaystyle A,}$  an involution ${\displaystyle \,*,}$  and a quadratic form ${\displaystyle N,}$  which is called the "norm". In several cases ${\displaystyle N}$  is an isotropic quadratic form so that ${\displaystyle A}$  has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.

An ultraseminorm or a non-Archimedean seminorm is a seminorm ${\displaystyle p:X\to \mathbb {R} }$  that also satisfies ${\displaystyle p(x+y)\leq \max\{p(x),p(y)\}{\text{ for all }}x,y\in X.}$ 

Weakening subadditivity: Quasi-seminorms

A map ${\displaystyle p:X\to \mathbb {R} }$  is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some ${\displaystyle b\leq 1}$  such that ${\displaystyle p(x+y)\leq bp(p(x)+p(y)){\text{ for all }}x,y\in X.}$  The smallest value of ${\displaystyle b}$  for which this holds is called the multiplier of ${\displaystyle p.}$ 

A quasi-seminorm that separates points is called a quasi-norm on ${\displaystyle X.}$ 

Weakening homogeneity - ${\displaystyle k}$ -seminorms

A map ${\displaystyle p:X\to \mathbb {R} }$  is called a ${\displaystyle k}$ -seminorm if it is subadditive and there exists a ${\displaystyle k}$  such that ${\displaystyle 0<k\leq 1}$  and for all ${\displaystyle x\in X}$  and scalars ${\displaystyle s,}$ 

$${\displaystyle p(sx)=|s|^{k}p(x)}$$

 

A ${\displaystyle k}$ -seminorm that separates points is called a ${\displaystyle k}$ -norm on ${\displaystyle X.}$ 

We have the following relationship between quasi-seminorms and ${\displaystyle k}$ -seminorms:

Suppose that ${\displaystyle q}$  is a quasi-seminorm on a vector space ${\displaystyle X}$  with multiplier ${\displaystyle b.}$  If ${\displaystyle 0<{\sqrt {k}}<\log _{2}b}$  then there exists ${\displaystyle k}$ -seminorm ${\displaystyle p}$  on ${\displaystyle X}$  equivalent to ${\displaystyle q.}$ 

See also

• Asymmetric norm – Generalization of the concept of a norm
• Banach space – Normed vector space that is complete
• Contraction mapping – Function reducing distance between all points
• Finest locally convex topology – A vector space with a topology defined by convex open setsPages displaying short descriptions of redirect targets
• Hahn-Banach theorem – Theorem on extension of bounded linear functionalsPages displaying short descriptions of redirect targets
• Gowers norm
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Mahalanobis distance – Statistical distance measure
• Matrix norm – Norm on a vector space of matrices
• Minkowski functional – Function made from a set
• Norm (mathematics) – Length in a vector space
• Normed vector space – Vector space on which a distance is defined
• Relation of norms and metrics – Mathematical space with a notion of distancePages displaying short descriptions of redirect targets
• Sublinear function – Type of function in linear algebra

Notes

Proofs

1. If ${\displaystyle z\in X}$  denotes the zero vector in ${\displaystyle X}$  while ${\displaystyle 0}$  denote the zero scalar, then absolute homogeneity implies that ${\displaystyle p(0)=p(0z)=|0|p(z)=0p(z)=0.}$  ${\displaystyle \blacksquare }$ 
2. Suppose ${\displaystyle p:X\to \mathbb {R} }$  is a seminorm and let ${\displaystyle x\in X.}$  Then absolute homogeneity implies ${\displaystyle p(-x)=p((-1)x)=|-1|p(x)=p(x).}$  The triangle inequality now implies ${\displaystyle p(0)=p(x+(-x))\leq p(x)+p(-x)=p(x)+p(x)=2p(x).}$  Because ${\displaystyle x}$  was an arbitrary vector in ${\displaystyle X,}$  it follows that ${\displaystyle p(0)\leq 2p(0),}$  which implies that ${\displaystyle 0\leq p(0)}$  (by subtracting ${\displaystyle p(0)}$  from both sides). Thus ${\displaystyle 0\leq p(0)\leq 2p(x)}$  which implies ${\displaystyle 0\leq p(x)}$  (by multiplying thru by ${\displaystyle 1/2}$ ).
3. Let ${\displaystyle x\in X}$  and ${\displaystyle k\in p^{-1}(0).}$  It remains to show that ${\displaystyle p(x+k)=p(x).}$  The triangle inequality implies ${\displaystyle p(x+k)\leq p(x)+p(k)=p(x)+0=p(x).}$  Since ${\displaystyle p(-k)=0,}$  ${\displaystyle p(x)=p(x)-p(-k)\leq p(x-(-k))=p(x+k),}$  as desired. ${\displaystyle \blacksquare }$ 

References

1. Kubrusly 2011, p. 200.
2. Narici & Beckenstein 2011, pp. 120–121.
3. Narici & Beckenstein 2011, pp. 116–128.
4. Wilansky 2013, pp. 15–21.
5. Schaefer & Wolff 1999, p. 40.
6. Narici & Beckenstein 2011, pp. 177–220.
7. Narici & Beckenstein 2011, pp. 116−128.
8. Narici & Beckenstein 2011, pp. 107–113.
9. Schechter 1996, p. 691.
10. Narici & Beckenstein 2011, p. 149.
11. Narici & Beckenstein 2011, pp. 149–153.
12. Wilansky 2013, pp. 18–21.
13. Obvious if ${\displaystyle X}$  is a real vector space. For the non-trivial direction, assume that ${\displaystyle \operatorname {Re} f\leq p}$  on ${\displaystyle X}$  and let

April 12, 2024