Quasinorm (original) (raw)

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In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by ‖ x + y ‖ ≤ K ( ‖ x ‖ + ‖ y ‖ ) {\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)} {\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)}for some K > 1. {\displaystyle K>1.} {\displaystyle K>1.}

A quasi-seminorm[1] on a vector space X {\displaystyle X} {\displaystyle X} is a real-valued map p {\displaystyle p} {\displaystyle p} on X {\displaystyle X} {\displaystyle X} that satisfies the following conditions:

  1. Non-negativity: p ≥ 0 ; {\displaystyle p\geq 0;} {\displaystyle p\geq 0;}
  2. Absolute homogeneity: 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;}
  3. there exists a real k ≥ 1 {\displaystyle k\geq 1} {\displaystyle k\geq 1} such that p ( x + y ) ≤ k [ p ( x ) + p ( y ) ] {\displaystyle p(x+y)\leq k[p(x)+p(y)]} {\displaystyle p(x+y)\leq k[p(x)+p(y)]} for all x , y ∈ X . {\displaystyle x,y\in X.} {\displaystyle x,y\in X.}
    • If k = 1 {\displaystyle k=1} {\displaystyle k=1} then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.

A quasinorm[1] is a quasi-seminorm that also satisfies:

  1. Positive definite/Point-separating: if 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 pair ( X , p ) {\displaystyle (X,p)} {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} {\displaystyle X} and an associated quasi-seminorm p {\displaystyle p} {\displaystyle p} is called a quasi-seminormed vector space. If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space.

Multiplier

The infimum of all values of k {\displaystyle k} {\displaystyle k} that satisfy condition (3) is called the multiplier of p . {\displaystyle p.} {\displaystyle p.}The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term k {\displaystyle k} {\displaystyle k}-quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to k . {\displaystyle k.} {\displaystyle k.}

A norm (respectively, a seminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is 1. {\displaystyle 1.} {\displaystyle 1.}Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).

If p {\displaystyle p} {\displaystyle p} is a quasinorm on X {\displaystyle X} {\displaystyle X} then p {\displaystyle p} {\displaystyle p} induces a vector topology on X {\displaystyle X} {\displaystyle X} whose neighborhood basis at the origin is given by the sets:[2] { x ∈ X : p ( x ) < 1 / n } {\displaystyle \{x\in X:p(x)<1/n\}} {\displaystyle \{x\in X:p(x)<1/n\}}as n {\displaystyle n} {\displaystyle n} ranges over the positive integers. A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space.

Every quasinormed topological vector space is pseudometrizable.

A complete quasinormed space is called a quasi-Banach space. Every Banach space is a quasi-Banach space, although not conversely.

A quasinormed space ( A , ‖ ⋅ ‖ ) {\displaystyle (A,\|\,\cdot \,\|)} {\displaystyle (A,\|\,\cdot \,\|)} is called a quasinormed algebra if the vector space A {\displaystyle A} {\displaystyle A} is an algebra and there is a constant K > 0 {\displaystyle K>0} {\displaystyle K>0} such that ‖ x y ‖ ≤ K ‖ x ‖ ⋅ ‖ y ‖ {\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|} {\displaystyle \|xy\|\leq K\|x\|\cdot \|y\|}for all x , y ∈ A . {\displaystyle x,y\in A.} {\displaystyle x,y\in A.}

A complete quasinormed algebra is called a quasi-Banach algebra.

A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.[2]

Since every norm is a quasinorm, every normed space is also a quasinormed space.

L p {\displaystyle L^{p}} {\displaystyle L^{p}} spaces with 0 < p < 1 {\displaystyle 0<p<1} {\displaystyle 0<p<1}

The L p {\displaystyle L^{p}} {\displaystyle L^{p}} spaces for 0 < p < 1 {\displaystyle 0<p<1} {\displaystyle 0<p<1} are quasinormed spaces (indeed, they are even F-spaces) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology). For 0 < p < 1 , {\displaystyle 0<p<1,} {\displaystyle 0<p<1,} the Lebesgue space L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} {\displaystyle L^{p}([0,1])} is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} {\displaystyle L^{p}([0,1])} and the empty set) and the only continuous linear functional on L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} {\displaystyle L^{p}([0,1])} is the constant 0 {\displaystyle 0} {\displaystyle 0} function (Rudin 1991, §1.47). In particular, the Hahn-Banach theorem does not hold for L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} {\displaystyle L^{p}([0,1])} when 0 < p < 1. {\displaystyle 0<p<1.} {\displaystyle 0<p<1.}

  1. ^ a b Kalton 1986, pp. 297–324.
  2. ^ a b Wilansky 2013, p. 55.