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

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.

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 ${\displaystyle rq\leq p\leq Rq.}$

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

References

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External links

• Sublinear functions
• The sandwich theorem for sublinear and super linear functionals