seminorm


Let V be a real, or a complex vector space, with K denoting the corresponding field of scalars. A seminorm is a functionMathworldPlanetmath

p:V+,

from V to the set of non-negative real numbers, that satisfies the following two properties.

p(k𝐮) =|k|p(𝐮),kK,𝐮V Homogeneity
p(𝐮+𝐯) p(𝐮)+p(𝐯),𝐮,𝐯V, Sublinearity

A seminorm differs from a norm in that it is permitted that p(𝐮)=0 for some non-zero 𝐮V.

It is possible to characterize the seminorms properties geometrically. For k>0, let

Bk={𝐮V:p(𝐮)k}

denote the ball of radius k. The homogeneity property is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the assertion that

Bk=kB1,

in the sense that 𝐮B1 if and only if k𝐮Bk. Thus, we see that a seminorm is fully determined by its unit ball. Indeed, given BV we may define a function pB:V+ by

pB(𝐮)=inf{λ+:λ-1𝐮B}.

The geometric nature of the unit ball is described by the following.

Proposition 1

The function pB satisfies the homegeneity property if and only if for every uV, there exists a kR+{} such that

λ𝐮Bif and only ifλk.
Proposition 2

Suppose that p is homogeneousPlanetmathPlanetmathPlanetmathPlanetmath. Then, it is sublinear if and only if its unit ball, B1, is a convex subset of V.

Proof. First, let us suppose that the seminorm is both sublinear and homogeneous, and prove that B1 is necessarily convex. Let 𝐮,𝐯B1, and let k be a real number between 0 and 1. We must show that the weighted average k𝐮+(1-k)𝐯 is in B1 as well. By assumptionPlanetmathPlanetmath,

p(k𝐮+(1-k)𝐯)kp(𝐮)+(1-k)p(𝐯).

The right side is a weighted average of two numbers between 0 and 1, and is therefore between 0 and 1 itself. Therefore

k𝐮+(1-k)𝐯B1,

as desired.

Conversely, suppose that the seminorm function is homogeneous, and that the unit ball is convex. Let 𝐮,𝐯V be given, and let us show that

p(𝐮+𝐯)p(𝐮)+p(𝐯).

The essential complication here is that we do not exclude the possibility that p(𝐮)=0, but that 𝐮0. First, let us consider the case where

p(𝐮)=p(𝐯)=0.

By homogeneity, for every k>0 we have

k𝐮,k𝐯B1,

and hence

k2𝐮+k2𝐯B1,

as well. By homogeneity, again,

p(𝐮+𝐯)2k.

Since the above is true for all positive k, we infer that

p(𝐮+𝐯)=0,

as desired.

Next suppose that p(𝐮)=0, but that p(𝐯)0. We will show that in this case, necessarily,

p(𝐮+𝐯)=p(𝐯).

Owing to the homogeneity assumption, we may without loss of generality assume that

p(𝐯)=1.

For every k such that 0k<1 we have

k𝐮+k𝐯=(1-k)k𝐮1-k+k𝐯.

The right-side expression is an element of B1 because

k𝐮1-k,𝐯B1.

Hence

kp(𝐮+𝐯)1,

and since this holds for k arbitrarily close to 1 we conclude that

p(𝐮+𝐯)p(𝐯).

The same argumentMathworldPlanetmath also shows that

p(𝐯)=p(-𝐮+(𝐮+𝐯))p(𝐮+𝐯),

and hence

p(𝐮+𝐯)=p(𝐯),

as desired.

Finally, suppose that neither p(𝐮) nor p(𝐯) is zero. Hence,

𝐮p(u),𝐯p(v)

are both in B1, and hence

p(u)p(u)+p(v)𝐮p(u)+p(v)p(u)+p(v)𝐯p(v)=𝐮+𝐯p(u)+p(v)

is in B1 also. Using homogeneity, we conclude that

p(u+v)p(u)+p(v),

as desired.

Title seminorm
Canonical name Seminorm
Date of creation 2013-03-22 12:24:57
Last modified on 2013-03-22 12:24:57
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 20
Author rmilson (146)
Entry type Definition
Classification msc 46B20
Synonym semi-norm
Defines homogeneous