Seminorm

Mathematical function

In mathematics, particularly in functional analysis, a seminorm is like a norm but 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

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

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

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

3. Nonnegativity:[1] p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all  x ∈ X . {\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  X {\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  x ∈ X {\displaystyle x\in X} satisfies  p ( x ) = 0 , {\displaystyle p(x)=0,} then  x = 0. {\displaystyle x=0.}

A seminormed space is a pair  ( X , p ) {\displaystyle (X,p)} consisting of a vector space  X {\displaystyle X} and a seminorm  p {\displaystyle p} on  X . {\displaystyle X.} If the seminorm  p {\displaystyle p} is also a norm then the seminormed space  ( X , p ) {\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  p : X -> R {\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  p : X -> R {\displaystyle p:X\to \mathbb {R} } is a seminorm if and only if it is a sublinear and balanced function.

## Examples

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* The trivial seminorm on  X , {\displaystyle X,} which refers to the constant  0 {\displaystyle 0} map on  X , {\displaystyle X,} induces the indiscrete topology on  X . {\displaystyle X.}
* Let  μ {\displaystyle \mu } be a measure on a space  Ω {\displaystyle \Omega } . For an arbitrary constant  c ≥ 1 {\displaystyle c\geq 1} , let  X {\displaystyle X} be the set of all functions  f : Ω -> R {\displaystyle f:\Omega \rightarrow \mathbb {R} } for which  ‖ f ‖ c := ( ∫ Ω | f | c d μ ) 1 / c {\displaystyle \lVert f\rVert _{c}:=\left(\int _{\Omega }|f|^{c}\,d\mu \right)^{1/c}} exists and is finite. It can be shown that  X {\displaystyle X} is a vector space, and the functional  ‖ ⋅ ‖ c {\displaystyle \lVert \cdot \rVert _{c}} is a seminorm on  X {\displaystyle X} . However, it is not always a norm (e.g. if  Ω = R {\displaystyle \Omega =\mathbb {R} } and  μ {\displaystyle \mu } is the Lebesgue measure) because  ‖ h ‖ c = 0 {\displaystyle \lVert h\rVert _{c}=0} does not always imply  h = 0 {\displaystyle h=0} . To make  ‖ ⋅ ‖ c {\displaystyle \lVert \cdot \rVert _{c}} a norm, quotient  X {\displaystyle X} by the closed subspace of functions  h {\displaystyle h} with  ‖ h ‖ c = 0 {\displaystyle \lVert h\rVert _{c}=0} . The resulting space,  L c ( μ ) {\displaystyle L^{c}(\mu )} , has a norm induced by  ‖ ⋅ ‖ c {\displaystyle \lVert \cdot \rVert _{c}} .
* If  f {\displaystyle f} is any linear form on a vector space then its absolute value | f | , {\displaystyle |f|,} defined by  x ↦ | f ( x ) | , {\displaystyle x\mapsto |f(x)|,} is a seminorm.
* A sublinear function f : X -> R {\displaystyle f:X\to \mathbb {R} } on a real vector space  X {\displaystyle X} is a seminorm if and only if it is a symmetric function, meaning that  f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for all  x ∈ X . {\displaystyle x\in X.}
* Every real-valued sublinear function f : X -> R {\displaystyle f:X\to \mathbb {R} } on a real vector space  X {\displaystyle X} induces a seminorm  p : X -> R {\displaystyle p:X\to \mathbb {R} } defined by  p ( x ) := max { f ( x ) , f ( − x ) } . {\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  p : X -> R {\displaystyle p:X\to \mathbb {R} } and  q : Y -> R {\displaystyle q:Y\to \mathbb {R} } are seminorms (respectively, norms) on  X {\displaystyle X} and  Y {\displaystyle Y} then the map  r : X × Y -> R {\displaystyle r:X\times Y\to \mathbb {R} } defined by  r ( x , y ) = p ( x ) + q ( y ) {\displaystyle r(x,y)=p(x)+q(y)} is a seminorm (respectively, a norm) on  X × Y . {\displaystyle X\times Y.} In particular, the maps on  X × Y {\displaystyle X\times Y} defined by  ( x , y ) ↦ p ( x ) {\displaystyle (x,y)\mapsto p(x)} and  ( x , y ) ↦ q ( y ) {\displaystyle (x,y)\mapsto q(y)} are both seminorms on  X × Y . {\displaystyle X\times Y.}
* If  p {\displaystyle p} and  q {\displaystyle q} are seminorms on  X {\displaystyle X} then so are[3] ( p ∨ q ) ( x ) = max { p ( x ) , q ( x ) } {\displaystyle (p\vee q)(x)=\max\\{p(x),q(x)\\}} and  ( p ∧ q ) ( x ) := inf { p ( y ) + q ( z ) : x = y + z with y , z ∈ X } {\displaystyle (p\wedge q)(x):=\inf\\{p(y)+q(z):x=y+z{\text{ with }}y,z\in X\\}} where  p ∧ q ≤ p {\displaystyle p\wedge q\leq p} and  p ∧ q ≤ q . {\displaystyle p\wedge q\leq q.} [4]
* The space of seminorms on  X {\displaystyle X} is generally not a distributive lattice with respect to the above operations. For example, over  R 2 {\displaystyle \mathbb {R} ^{2}} ,  p ( x , y ) := max ( | x | , | y | ) , q ( x , y ) := 2 | x | , r ( x , y ) := 2 | y | {\displaystyle p(x,y):=\max(|x|,|y|),q(x,y):=2|x|,r(x,y):=2|y|} are such that  ( ( p ∨ q ) ∧ ( p ∨ r ) ) ( x , y ) = inf { max ( 2 | x 1 | , | y 1 | ) + max ( | x 2 | , 2 | y 2 | ) : x = x 1 + x 2 and y = y 1 + y 2 } {\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  ( p ∨ q ∧ r ) ( x , y ) := max ( | x | , | y | ) {\displaystyle (p\vee q\wedge r)(x,y):=\max(|x|,|y|)}
* If  L : X -> Y {\displaystyle L:X\to Y} is a linear map and  q : Y -> R {\displaystyle q:Y\to \mathbb {R} } is a seminorm on  Y , {\displaystyle Y,} then  q ∘ L : X -> R {\displaystyle q\circ L:X\to \mathbb {R} } is a seminorm on  X . {\displaystyle X.} The seminorm  q ∘ L {\displaystyle q\circ L} will be a norm on  X {\displaystyle X} if and only if  L {\displaystyle L} is injective and the restriction  q | L ( X ) {\displaystyle q{\big \vert }_{L(X)}} is a norm on  L ( X ) . {\displaystyle L(X).}

## Minkowski functionals and seminorms

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Main article: Minkowski functional

Seminorms on a vector space  X {\displaystyle X} are intimately tied, via Minkowski functionals, to subsets of  X {\displaystyle X} that are convex, balanced, and absorbing. Given such a subset  D {\displaystyle D} of  X , {\displaystyle X,} the Minkowski functional of  D {\displaystyle D} is a seminorm. Conversely, given a seminorm  p {\displaystyle p} on  X , {\displaystyle X,} the sets { x ∈ X : p ( x ) < 1 } {\displaystyle \\{x\in X:p(x)<1\\}} and  { x ∈ X : p ( x ) ≤ 1 } {\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  p . {\displaystyle p.} [5]

## Algebraic properties

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Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity,  p ( 0 ) = 0 , {\displaystyle p(0)=0,} and for all vectors  x , y ∈ X {\displaystyle x,y\in X} : the reverse triangle inequality: [2][6] | p ( x ) − p ( y ) | ≤ p ( x − y ) {\displaystyle |p(x)-p(y)|\leq p(x-y)} and also  0 ≤ max { p ( x ) , p ( − x ) } {\textstyle 0\leq \max\\{p(x),p(-x)\\}} and  p ( x ) − p ( y ) ≤ p ( x − y ) . {\displaystyle p(x)-p(y)\leq p(x-y).} [2][6]

For any vector  x ∈ X {\displaystyle x\in X} and positive real  r > 0 : {\displaystyle r>0:} [7] x + { y ∈ X : p ( y ) < r } = { y ∈ X : p ( x − y ) < r } {\displaystyle x+\\{y\in X:p(y)<r\\}=\\{y\in X:p(x-y)<r\\}} and furthermore,  { x ∈ X : p ( x ) < r } {\displaystyle \\{x\in X:p(x)<r\\}} is an absorbing disk in  X . {\displaystyle X.} [3]

If  p {\displaystyle p} is a sublinear function on a real vector space  X {\displaystyle X} then there exists a linear functional  f {\displaystyle f} on  X {\displaystyle X} such that  f ≤ p {\displaystyle f\leq p} [6] and furthermore, for any linear functional  g {\displaystyle g} on  X , {\displaystyle X,} g ≤ p {\displaystyle g\leq p} on  X {\displaystyle X} if and only if  g − 1 ( 1 ) ∩ { x ∈ X : p ( x ) < 1 } = ∅ . {\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  p {\displaystyle p} is a norm on  X {\displaystyle X} if and only if  { x ∈ X : p ( x ) < 1 } {\displaystyle \\{x\in X:p(x)<1\\}} does not contain a non-trivial vector subspace.

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

Furthermore, for any real  r > 0 , {\displaystyle r>0,} [3] r { x ∈ X : p ( x ) < 1 } = { x ∈ X : p ( x ) < r } = { x ∈ X : 1 r p ( x ) < 1 } . {\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  D {\displaystyle D} is a set satisfying  { x ∈ X : p ( x ) < 1 } ⊆ D ⊆ { x ∈ X : p ( x ) ≤ 1 } {\displaystyle \\{x\in X:p(x)<1\\}\subseteq D\subseteq \\{x\in X:p(x)\leq 1\\}} then  D {\displaystyle D} is absorbing in  X {\displaystyle X} and  p = p D {\displaystyle p=p_{D}} where  p D {\displaystyle p_{D}} denotes the Minkowski functional associated with  D {\displaystyle D} (that is, the gauge of  D {\displaystyle D} ).[5] In particular, if  D {\displaystyle D} is as above and  q {\displaystyle q} is any seminorm on  X , {\displaystyle X,} then  q = p {\displaystyle q=p} if and only if  { x ∈ X : q ( x ) < 1 } ⊆ D ⊆ { x ∈ X : q ( x ) ≤ } . {\displaystyle \\{x\in X:q(x)<1\\}\subseteq D\subseteq \\{x\in X:q(x)\leq \\}.} [5]

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

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

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

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

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

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

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

### Inequalities involving seminorms

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

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

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

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

### Hahn–Banach theorem for seminorms

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

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

A similar extension property also holds for seminorms:

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

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

## Topologies of seminormed spaces

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### Pseudometrics and the induced topology

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A seminorm  p {\displaystyle p} on  X {\displaystyle X} induces a topology, called the seminorm-induced topology, via the canonical translation-invariant pseudometric d p : X × X -> R {\displaystyle d_{p}:X\times X\to \mathbb {R} } ;  d p ( x , y ) := p ( x − y ) = p ( y − x ) . {\displaystyle d_{p}(x,y):=p(x-y)=p(y-x).} This topology is Hausdorff if and only if  d p {\displaystyle d_{p}} is a metric, which occurs if and only if  p {\displaystyle p} is a norm.[4] This topology makes  X {\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:  { x ∈ X : p ( x ) < r } or { x ∈ X : p ( x ) ≤ r } {\displaystyle \\{x\in X:p(x)<r\\}\quad {\text{ or }}\quad \\{x\in X:p(x)\leq r\\}} as  r > 0 {\displaystyle r>0} ranges over the positive reals. Every seminormed space  ( X , p ) {\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  X {\displaystyle X} with seminorm  p {\displaystyle p} induces a vector space quotient X / W , {\displaystyle X/W,} where  W {\displaystyle W} is the subspace of  X {\displaystyle X} consisting of all vectors  x ∈ X {\displaystyle x\in X} with  p ( x ) = 0. {\displaystyle p(x)=0.} Then  X / W {\displaystyle X/W} carries a norm defined by  p ( x + W ) = p ( x ) . {\displaystyle p(x+W)=p(x).} The resulting topology, pulled back to  X , {\displaystyle X,} is precisely the topology induced by  p . {\displaystyle p.}

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

#### Stronger, weaker, and equivalent seminorms

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

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

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

### Normability and seminormability

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

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

Furthermore,  X {\displaystyle X} is finite dimensional if and only if  X σ ′ {\displaystyle X_{\sigma }^{\prime }} is normable (here  X σ ′ {\displaystyle X_{\sigma }^{\prime }} denotes  X ′ {\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

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* If  X {\displaystyle X} is a TVS and  p {\displaystyle p} is a continuous seminorm on  X , {\displaystyle X,} then the closure of  { x ∈ X : p ( x ) < r } {\displaystyle \\{x\in X:p(x)<r\\}} in  X {\displaystyle X} is equal to  { x ∈ X : p ( x ) ≤ r } . {\displaystyle \\{x\in X:p(x)\leq r\\}.} [3]
* The closure of  { 0 } {\displaystyle \\{0\\}} in a locally convex space  X {\displaystyle X} whose topology is defined by a family of continuous seminorms  P {\displaystyle {\mathcal {P}}} is equal to  ⋂ p ∈ P p − 1 ( 0 ) . {\displaystyle \bigcap _{p\in {\mathcal {P}}}p^{-1}(0).} [11]
* A subset  S {\displaystyle S} in a seminormed space  ( X , p ) {\displaystyle (X,p)} is bounded if and only if  p ( S ) {\displaystyle p(S)} is bounded.[20]
* If  ( X , p ) {\displaystyle (X,p)} is a seminormed space then the locally convex topology that  p {\displaystyle p} induces on  X {\displaystyle X} makes  X {\displaystyle X} into a pseudometrizable TVS with a canonical pseudometric given by  d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)} for all  x , y ∈ X . {\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

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

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

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

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

### Continuity of linear maps

[edit]

If  F : ( X , p ) -> ( Y , q ) {\displaystyle F:(X,p)\to (Y,q)} is a map between seminormed spaces then let[15] ‖ F ‖ p , q := sup { q ( F ( x ) ) : p ( x ) ≤ 1 , x ∈ X } . {\displaystyle \|F\|_{p,q}:=\sup\\{q(F(x)):p(x)\leq 1,x\in X\\}.}

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

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

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

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

## Generalizations

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

A composition algebra  ( A , ∗ , N ) {\displaystyle (A,*,N)} consists of an algebra over a field A , {\displaystyle A,} an involution ∗ , {\displaystyle \,*,} and a quadratic form N , {\displaystyle N,} which is called the "norm". In several cases  N {\displaystyle N} is an isotropic quadratic form so that  A {\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  p : X -> R {\displaystyle p:X\to \mathbb {R} } that also satisfies  p ( x + y ) ≤ max { p ( x ) , p ( y ) } for all x , y ∈ X . {\displaystyle p(x+y)\leq \max\\{p(x),p(y)\\}{\text{ for all }}x,y\in X.}

Weakening subadditivity: Quasi-seminorms

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

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

Weakening homogeneity -  k {\displaystyle k} -seminorms

A map  p : X -> R {\displaystyle p:X\to \mathbb {R} } is called a  k {\displaystyle k} -seminorm if it is subadditive and there exists a  k {\displaystyle k} such that  0 < k ≤ 1 {\displaystyle 0<k\leq 1} and for all  x ∈ X {\displaystyle x\in X} and scalars  s , {\displaystyle s,} p ( s x ) = | s | k p ( x ) {\displaystyle p(sx)=|s|^{k}p(x)} A  k {\displaystyle k} -seminorm that separates points is called a  k {\displaystyle k} -norm on  X . {\displaystyle X.}

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

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

## See also

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* 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 – 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 – Class of norms in additive combinatorics
* Locally convex topological vector space – 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

[edit]

Proofs

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

## References

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1. ^ a b c d Kubrusly 2011, p. 200.
2. ^ a b c Narici & Beckenstein 2011, pp. 120–121.
3. ^ a b c d e f g h i j Narici & Beckenstein 2011, pp. 116–128.
4. ^ a b c d e f g Wilansky 2013, pp. 15–21.
5. ^ a b c d Schaefer & Wolff 1999, p. 40.
6. ^ a b c d e f g 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  X {\displaystyle X} is a real vector space. For the non-trivial direction, assume that  Re ⁡ f ≤ p {\displaystyle \operatorname {Re} f\leq p} on  X {\displaystyle X} and let  x ∈ X . {\displaystyle x\in X.} Let  r ≥ 0 {\displaystyle r\geq 0} and  t {\displaystyle t} be real numbers such that  f ( x ) = r e i t . {\displaystyle f(x)=re^{it}.} Then  | f ( x ) | = r = f ( e − i t x ) = Re ⁡ ( f ( e − i t x ) ) ≤ p ( e − i t x ) = p ( x ) . {\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. ^ a b c d e f 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.

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