Seminorm

☆ Save On Wikipedia ↗

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

Let X {\displaystyle X} {\displaystyle X} be a vector space over either the real numbers R {\displaystyle \mathbb {R} } {\displaystyle \mathbb {R} } or the complex numbers C . {\displaystyle \mathbb {C} .} {\displaystyle \mathbb {C} .} A real-valued function p : X → R {\displaystyle p:X\to \mathbb {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)} {\displaystyle p(x+y)\leq p(x)+p(y)} for all x , y ∈ X . {\displaystyle x,y\in X.} {\displaystyle x,y\in X.}
  2. Absolute homogeneity:[1] p ( s x ) = | s | p ( x ) {\displaystyle p(sx)=|s|p(x)} {\displaystyle p(sx)=|s|p(x)} for all x ∈ X {\displaystyle x\in X} {\displaystyle x\in X} and all scalars s . {\displaystyle s.} {\displaystyle s.}

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

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

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

Minkowski functionals and seminorms

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

Algebraic properties

Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity, p ( 0 ) = 0 , {\displaystyle p(0)=0,} {\displaystyle p(0)=0,} and for all vectors x , y ∈ X {\displaystyle x,y\in 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)} {\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)\}} {\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).} {\displaystyle p(x)-p(y)\leq p(x-y).}[2][6]

For any vector x ∈ X {\displaystyle x\in X} {\displaystyle x\in X} and positive real r > 0 : {\displaystyle 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\}} {\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\}} {\displaystyle \{x\in X:p(x)<r\}} is an absorbing disk in X . {\displaystyle X.} {\displaystyle X.}[3]

If p {\displaystyle p} {\displaystyle p} is a sublinear function on a real vector space X {\displaystyle X} {\displaystyle X} then there exists a linear functional f {\displaystyle f} {\displaystyle f} on X {\displaystyle X} {\displaystyle X} such that f ≤ p {\displaystyle f\leq p} {\displaystyle f\leq p}[6] and furthermore, for any linear functional g {\displaystyle g} {\displaystyle g} on X , {\displaystyle X,} {\displaystyle X,} g ≤ p {\displaystyle g\leq p} {\displaystyle g\leq p} on X {\displaystyle 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 .} {\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} {\displaystyle p} is a norm on X {\displaystyle X} {\displaystyle X} if and only if { x ∈ X : p ( x ) < 1 } {\displaystyle \{x\in 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 )} {\displaystyle p:X\to [0,\infty )} is a seminorm on X {\displaystyle X} {\displaystyle X} then ker ⁡ p := p − 1 ( 0 ) {\displaystyle \ker p:=p^{-1}(0)} {\displaystyle \ker p:=p^{-1}(0)} is a vector subspace of X {\displaystyle X} {\displaystyle X} and for every x ∈ X , {\displaystyle x\in X,} {\displaystyle x\in X,} p {\displaystyle 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\}} {\displaystyle x+\ker p=\{x+k:p(k)=0\}} and equal to p ( x ) . {\displaystyle p(x).} {\displaystyle p(x).}[proof 3]

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

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

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

  1. p {\displaystyle p} {\displaystyle p} is a seminorm.
  2. p {\displaystyle p} {\displaystyle p} is a convex F {\displaystyle F} {\displaystyle F}-seminorm.
  3. p {\displaystyle 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} {\displaystyle p} is a norm;
  2. { x ∈ X : p ( x ) < 1 } {\displaystyle \{x\in 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,} {\displaystyle X,} with respect to which, { x ∈ X : p ( x ) < 1 } {\displaystyle \{x\in X:p(x)<1\}} {\displaystyle \{x\in X:p(x)<1\}} is bounded.

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

  1. p {\displaystyle 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} {\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} {\displaystyle p(x)+p(-x)=0{\text{ for every }}x\in X};

Inequalities involving seminorms

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

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

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

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

Hahn–Banach theorem for seminorms

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

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

A similar extension property also holds for seminorms:

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

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

Topologies of seminormed spaces

Pseudometrics and the induced topology

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

Any seminorm-induced topology makes X {\displaystyle X} {\displaystyle X} locally convex, as follows. If p {\displaystyle p} {\displaystyle p} is a seminorm on X {\displaystyle X} {\displaystyle X} and r ∈ R , {\displaystyle r\in \mathbb {R} ,} {\displaystyle r\in \mathbb {R} ,} call the set { x ∈ X : p ( x ) < r } {\displaystyle \{x\in X:p(x)<r\}} {\displaystyle \{x\in X:p(x)<r\}} the open ball of radius r {\displaystyle r} {\displaystyle r} about the origin; likewise the closed ball of radius r {\displaystyle r} {\displaystyle r} is { x ∈ X : p ( x ) ≤ r } . {\displaystyle \{x\in X:p(x)\leq r\}.} {\displaystyle \{x\in X:p(x)\leq r\}.} The set of all open (resp. closed) p {\displaystyle 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} {\displaystyle p}-topology on X . {\displaystyle X.} {\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 p {\displaystyle p} {\displaystyle p} and q {\displaystyle q} {\displaystyle q} are seminorms on X , {\displaystyle X,} {\displaystyle X,} then we say that q {\displaystyle q} {\displaystyle q} is stronger than p {\displaystyle p} {\displaystyle p} and that p {\displaystyle p} {\displaystyle p} is weaker than q {\displaystyle q} {\displaystyle q} if any of the following equivalent conditions holds:

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

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

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

Furthermore, X {\displaystyle X} {\displaystyle X} is finite dimensional if and only if X σ ′ {\displaystyle X_{\sigma }^{\prime }} {\displaystyle X_{\sigma }^{\prime }} is normable (here X σ ′ {\displaystyle X_{\sigma }^{\prime }} {\displaystyle X_{\sigma }^{\prime }} denotes X ′ {\displaystyle X^{\prime }} {\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 X {\displaystyle X} {\displaystyle X} is a TVS and p {\displaystyle p} {\displaystyle p} is a continuous seminorm on X , {\displaystyle X,} {\displaystyle X,} then the closure of { x ∈ X : p ( x ) < r } {\displaystyle \{x\in X:p(x)<r\}} {\displaystyle \{x\in X:p(x)<r\}} in X {\displaystyle X} {\displaystyle X} is equal to { x ∈ X : p ( x ) ≤ r } . {\displaystyle \{x\in X:p(x)\leq r\}.} {\displaystyle \{x\in X:p(x)\leq r\}.}[3]
  • The closure of { 0 } {\displaystyle \{0\}} {\displaystyle \{0\}} in a locally convex space X {\displaystyle X} {\displaystyle X} whose topology is defined by a family of continuous seminorms P {\displaystyle {\mathcal {P}}} {\displaystyle {\mathcal {P}}} is equal to ⋂ p ∈ P p − 1 ( 0 ) . {\displaystyle \bigcap _{p\in {\mathcal {P}}}p^{-1}(0).} {\displaystyle \bigcap _{p\in {\mathcal {P}}}p^{-1}(0).}[11]
  • A subset S {\displaystyle S} {\displaystyle S} in a seminormed space ( X , p ) {\displaystyle (X,p)} {\displaystyle (X,p)} is bounded if and only if p ( S ) {\displaystyle p(S)} {\displaystyle p(S)} is bounded.[20]
  • If ( X , p ) {\displaystyle (X,p)} {\displaystyle (X,p)} is a seminormed space then the locally convex topology that p {\displaystyle p} {\displaystyle p} induces on X {\displaystyle X} {\displaystyle X} makes X {\displaystyle 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)} {\displaystyle d(x,y):=p(x-y)} for all x , y ∈ X . {\displaystyle x,y\in 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

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

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

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

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

Continuity of linear maps

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

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

If F {\displaystyle F} {\displaystyle F} is continuous then q ( F ( x ) ) ≤ ‖ F ‖ p , q p ( x ) {\displaystyle q(F(x))\leq \|F\|_{p,q}p(x)} {\displaystyle q(F(x))\leq \|F\|_{p,q}p(x)} for all x ∈ X . {\displaystyle x\in 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)} {\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}.} {\displaystyle \|F\|_{p,q}.} This seminorm is a norm if q {\displaystyle q} {\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 ( A , ∗ , N ) {\displaystyle (A,*,N)} {\displaystyle (A,*,N)} consists of an algebra over a field A , {\displaystyle A,} {\displaystyle A,} an involution ∗ , {\displaystyle \,*,} {\displaystyle \,*,} and a quadratic form N , {\displaystyle N,} {\displaystyle N,} which is called the "norm". In several cases N {\displaystyle N} {\displaystyle N} is an isotropic quadratic form so that A {\displaystyle 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} } {\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.} {\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} } {\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} {\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.} {\displaystyle p(x+y)\leq bp(p(x)+p(y)){\text{ for all }}x,y\in X.} The smallest value of b {\displaystyle b} {\displaystyle b} for which this holds is called the multiplier of p . {\displaystyle p.} {\displaystyle p.}

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

Weakening homogeneity - k {\displaystyle k} {\displaystyle k}-seminorms

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

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

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

See also

Notes

Proofs

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

References

  1. Kubrusly 2011, p. 200.
  2. Narici & Beckenstein 2011, pp. 120–121.
  3. Narici & Beckenstein 2011, pp. 116–128.
  4. Wilansky 2013, pp. 15–21.
  5. Schaefer & Wolff 1999, p. 40.
  6. Narici & Beckenstein 2011, pp. 177–220.
  7. Narici & Beckenstein 2011, pp. 116−128.
  8. Narici & Beckenstein 2011, pp. 107–113.
  9. Schechter 1996, p. 691.
  10. Narici & Beckenstein 2011, p. 149.
  11. Narici & Beckenstein 2011, pp. 149–153.
  12. Wilansky 2013, pp. 18–21.
  13. Obvious if X {\displaystyle X} {\displaystyle X} is a real vector space. For the non-trivial direction, assume that Re ⁡ f ≤ p {\displaystyle \operatorname {Re} f\leq p} {\displaystyle \operatorname {Re} f\leq p} on X {\displaystyle X} {\displaystyle X} and let x ∈ X . {\displaystyle x\in X.} {\displaystyle x\in X.} Let r ≥ 0 {\displaystyle r\geq 0} {\displaystyle r\geq 0} and t {\displaystyle t} {\displaystyle t} be real numbers such that f ( x ) = r e i t . {\displaystyle f(x)=re^{it}.} {\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).} {\displaystyle |f(x)|=r=f\left(e^{-it}x\right)=\operatorname {Re} \left(f\left(e^{-it}x\right)\right)\leq p\left(e^{-it}x\right)=p(x).}
  14. Wilansky 2013, p. 20.
  15. Wilansky 2013, pp. 21–26.
  16. Narici & Beckenstein 2011, pp. 150.
  17. Wilansky 2013, pp. 50–51.
  18. Narici & Beckenstein 2011, pp. 156–175.
  19. Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
  20. Wilansky 2013, pp. 49–50.
  21. Narici & Beckenstein 2011, pp. 115–154.