hyperplane separation
Let be a vector space, and be any subspace
of linear functionals
on .
Impose on the weak topology generated by .
Theorem 1 (Hyperplane Separation Theorem I).
Given a weakly closed convex subset , and . there is such that
Proof.
The weak topology on can be generated by the semi-norms for . A subbasis for the weak topology consists of neigborhoods of the form for , and . Since is weakly open, there exist and such that
In other words, if then at least one of is .
Define a map
by .
The set is evidently
closed and convex in , a Hilbert space
under the standard inner product
.
So there is a point
that minimizes the norm .
It follows that for all ; for otherwise we can attain a smaller value of the norm by moving from the point along a line towards . (Formally, we have .)
Take where . Then we find, for all ,
Theorem 2 (Hyperplane Separation Theorem II).
Let be a weakly closed convex subset, and a compact convex subset, that do not intersect each other. Then there exists such that
Proof.
We show that is weakly closed in . Let be a net convergent to . Since is compact, has a subnet convergent to . Then the subnet is convergent to . The point is in since is closed; therefore is in .
Also, is convex since and are. Noting that (otherwise and would have a common point), we apply the previous theorem to obtain a such that
The desired conclusion 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 |