BREAKING NEWS
Seminorm

## Summary

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]

1. 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:

1. 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)

and furthermore, ${\displaystyle \{x\in X:p(x)  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)

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)  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)\}  then ${\displaystyle q(x)  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)  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)

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)  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)  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 ${\displaystyle p(x+y)\leq \max\{p(x),p(y)\}{\text{ for all }}x,y\in X.}$

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  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 ${\displaystyle q.}$

## 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. ^ a b c d Kubrusly 2011, p. 200.
2. ^ a b c Narici & Beckenstein 2011, pp. 120–121.
3. Narici & Beckenstein 2011, pp. 116–128.
4. Wilansky 2013, pp. 15–21.
5. ^ a b c d 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. ^ a b Narici & Beckenstein 2011, p. 149.
11. ^ a b c d Narici & Beckenstein 2011, pp. 149–153.
12. ^ a b c 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. ^ a b c Narici & Beckenstein 2011, pp. 156–175.
19. ^ a b 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.
• Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
• Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401.
• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
• Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
• Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Kubrusly, Carlos S. (2011). The Elements of Operator Theory (Second ed.). Boston: Birkhäuser. ISBN 978-0-8176-4998-2. OCLC 710154895.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Prugovečki, Eduard (1981). Quantum mechanics in Hilbert space (2nd ed.). Academic Press. p. 20. ISBN 0-12-566060-X.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.