Hahn–Banach theorem

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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 Hahn–Banach theorem is known as 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, though it was proved earlier (in 1912) by Eduard Helly.[1]



The most general formulation of the theorem needs some preparation. Given a vector space V over the field R of real numbers, a function ƒ : VR is called sublinear if

Every seminorm on V (in particular, every norm on V) is sublinear. Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets.

The Hahn–Banach theorem states that if \scriptstyle\mathcal{N}:\; V\rightarrow\mathbb{R} is a sublinear function, and \scriptstyle\varphi:\; U\rightarrow\mathbb{R} is a linear functional on a linear subspace U of V which is dominated by \scriptstyle\mathcal{N} on U,

Another version of Hahn–Banach theorem states that if V is a vector space over the scalar field K (either the real numbers R or the complex numbers C), if \scriptstyle\mathcal{N}:\;V\rightarrow\mathbb{R} is a seminorm, and \scriptstyle\varphi:\;U\rightarrow\mathbb{K} is a K-linear functional on a K-linear subspace U of V which is dominated by \scriptstyle\mathcal{N} on U in absolute value,

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