The first geometric form of the Hahn–Banach theorem: in a separable real normed space, two disjoint nonempty convex sets, one of them open, are separated by a closed hyperplane F = α, with strict inequality on the open set.
In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let be a separable real normed space, and let be nonempty, disjoint and convex, with open.
Separation. There are an element of the dual space , not identically zero, and a real number with
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