hyperplane separation

Let X be a vector spaceMathworldPlanetmath, and Φ be any subspacePlanetmathPlanetmathPlanetmath of linear functionalsMathworldPlanetmathPlanetmath on X. Impose on X the weak topology generated by Φ.

Theorem 1 (Hyperplane Separation Theorem I).

Given a weakly closed convex subset SX, and aXS. there is ϕΦ such that


The weak topology on X can be generated by the semi-norms x|p(x)| for pΦ. A subbasis for the weak topology consists of neigborhoods of the form {xX:|p(x-y)|<ϵ} for yX, pΦ and ϵ>0. Since XS is weakly open, there exist f1,,fnΦ and ϵ>0 such that

|fi(x)-fi(a)|=|fi(x-a)|<ϵ, for all i=1,,n implies xXS.

In other words, if xS then at least one of |fi(x)-fi(a)| is ϵ.

Define a map F:Xn by F(x)=(f1(x),,fn(x)). The set F(S)¯ is evidently closed and convex in n, a Hilbert spaceMathworldPlanetmath under the standard inner productMathworldPlanetmath. So there is a point bF(S)¯ that minimizes the norm b-F(a).

It follows that y-b,b-F(a)0 for all yF(S)¯; for otherwise we can attain a smaller value of the norm by moving from the point b along a line towards y. (Formally, we have 0ddt|t=0ty+(1-t)b-F(a)2=2y-b,b-F(a).)

Take ϕ=i=1nλifi where λ=b-F(a). Then we find, for all xS,

ϕ(x-a) =b-F(a),F(x-a)
Theorem 2 (Hyperplane Separation Theorem II).

Let SX be a weakly closed convex subset, and KX a compact convex subset, that do not intersect each other. Then there exists ϕΦ such that


We show that S-K={x-y:xS,yK} is weakly closed in X. Let {zα=xα-yα}A be a net convergent to z. Since K is compact, {yα} has a subnet {yα(β)} convergent to yK. Then the subnet xα(β)=zα(β)+yα(β) is convergent to x=z+y. The point x is in S since S is closed; therefore z=x-y is in S-K.

Also, S-K is convex since S and K are. Noting that 0S-K (otherwise S and K would have a common point), we apply the previous theorem to obtain a ϕΦ such that

0=ϕ(0)<infzS-Kϕ(z)ϕ(x-y), for all xS and yK

The desired conclusionMathworldPlanetmath follows at once. ∎

Title hyperplane separation
Canonical name HyperplaneSeparation
Date of creation 2013-03-22 17:19:01
Last modified on 2013-03-22 17:19:01
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 4
Author stevecheng (10074)
Entry type Theorem
Classification msc 46A55
Classification msc 49J27
Classification msc 46A20
Synonym separating hyperplane
Related topic HahnBanachgeometricFormTheorem