Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the separating hyperplane theorem, and has numerous uses in convex geometry. It is named for Hans Hahn and Stefan Banach, who proved this theorem independently in the late 1920s, although a special case—for the space of continuous functions on an interval—was proved earlier (in 1912) by Eduard Helly, and a general extension theorem from which the Hahn–Banach theorem can be derived was proved in 1923 by Marcel Riesz.
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The most general formulation of the theorem needs some preparation. Given a real vector space V, a function f : V → R is called sublinear if
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