Separating hyperplane theorem (separation theorem in short) is an elementary result in functional analysis and convex analysis and is important to optimization. Here we only discuss the “weak version”. In Euclidean space we have the following result.
THEOREM 1. Let and be two nonempty convex disjoint subsets in . Then there exist and such that for any , . Note is the standard inner product on .
The proof of THEOREM 1 depends on the fact that any nonempty convex set in has nonempty relative interior. This is not the case if the underlying space is infinite dimensional.
It seems rare to see this general result in standard texts of functional analysis e.g.Rudin, what we can see is a weaker one:
THEOREM 2. Given a real topological vector space . Let and be two nonempty convex disjoint subsets in . If is open, then there exists and such that for any and .
The condition that one of , has nonempty interior cannot be dropped by considering the following example:
EXAMPLE 3 (The example and verification are not originated by me).
Consider . Let . Take to be , where .
It can be verified directly that is a closed convex set. Also and are disjoint since if , then for all which contradicts .
Now suppose and can be separated by some nontrivial continuous linear functional. Then and can also be separated by the same functional. If we can show is a dense subset of , then by limit arguments one can show the functional must be trivial, which is a contradiction.
Therefore, it suffices to show is a dense subset of . Indeed, let and be given. Choose a large positive integer such that . Then take a large number such that . Thus take . It follows from the choice of above that . Hence . The choice of above guarantees . We are done.
Clearly has empty interior. In fact also has empty interior. Indeed, pick and consider . Choose large such that and . Then one can check since under the choice of we have , where is defined above.
Remark: We can replace by , to construct a similar example.