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]

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

${\textstyle 0\leq \max\{p(x),p(-x)\))$

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

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\))$$

${\displaystyle \{x\in X:p(x)<r\))$

${\displaystyle X.}$

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:

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\))$$

${\displaystyle r>0}$

${\displaystyle (X,p)}$

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

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 ${\displaystyle s,}$

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

${\displaystyle k}$

${\displaystyle k}$-norm

${\displaystyle X.}$

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 ${\displaystyle q.}$