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 ${\displaystyle p(x+W)=p(x).}$ 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 ${\displaystyle x\in X.}$ Let ${\displaystyle r\geq 0}$ and ${\displaystyle t}$ be real numbers such that ${\displaystyle f(x)=re^{it}.}$ Then ${\displaystyle |f(x)|=r=f\left(e^{-it}x\right)=\operatorname {Re} \left(f\left(e^{-it}x\right)\right)\leq p\left(e^{-it}x\right)=p(x).}$
14. Wilansky 2013, p. 20.
15. Wilansky 2013, pp. 21–26.
16. Narici & Beckenstein 2011, pp. 150.
17. Wilansky 2013, pp. 50–51.
18. Narici & Beckenstein 2011, pp. 156–175.
19. Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
20. Wilansky 2013, pp. 49–50.
21. Narici & Beckenstein 2011, pp. 115–154.

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External links

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