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. 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. The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C [ a , b ] {displaystyle Cleft} of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. 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 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. Hahn–Banach theorem (Rudin 1991, Th. 3.2). If p : V → R is a sublinear function, and φ : U → R is a linear functional on a linear subspace U ⊆ V which is dominated by p on U, i.e. then there exists a linear extension ψ : V → R of φ to the whole space V, i.e., there exists a linear functional ψ such that Hahn–Banach theorem (alternative version). Set K = R or C and let V be a K-vector space with a seminorm p : V → R. If φ : U → K is a K-linear functional on a K-linear subspace U of V which is dominated by p on U in absolute value, then there exists a linear extension ψ : V → K of φ to the whole space V, i.e., there exists a K-linear functional ψ such that In the complex case of the alternate version, the C-linearity assumptions demand, in addition to the assumptions for the real case, that for every vector x ∈ U, we have ix ∈ U and φ(ix) = iφ(x).