Theorem 1 (Hyperplane Separation Theorem I).
Given a weakly closed convex subset , and . there is such that
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 .
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).
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 .
|Date of creation||2013-03-22 17:19:01|
|Last modified on||2013-03-22 17:19:01|
|Last modified by||stevecheng (10074)|