WebThe 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".

WebEn mathématiques, et plus particulièrement en analyse et en géométrie, le théorème de Hahn-Banach, dû aux deux mathématiciens Hans Hahn [1] et Stefan Banach [2], est un théorème d'existence de prolongements de formes linéaires satisfaisant à certaines conditions.. En permettant de prouver abstraitement l'existence de nombreuses fonctions …

WebThe Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.Indeed, the reassembly process …

WebThe Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence.The main condition is the equicontinuity of the family of …

WebIt is said that the graph of is closed if ⁡ is a closed subset of (with the product topology).. Any continuous function into a Hausdorff space has a closed graph.. Any linear map, :, between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) is sequentially continuous in the sense …

WebIn mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is …

WebOpen mapping theorem — Let : be a surjective linear map from an complete pseudometrizable TVS onto a TVS and suppose that at least one of the following two conditions is satisfied: . is a Baire space, or; is locally convex and is a barrelled space,; If is a closed linear operator then is an open mapping. If is a continuous linear operator and is …

WebIn calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration.This relationship is commonly characterized (by …

WebThe Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the optimally separating hyperplane or maximum-margin hyperplane is a hyperplane which separates two convex hulls of points and is equidistant from the two.

WebIn mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.. Statement of the theorem. Let and be Hilbert spaces, and let : ⁡ be an unbounded operator from into . Suppose that is a closed operator and that is densely defined, that is, ⁡ is dense in . Let : ⁡ denote the adjoint of . Then is also densely defined, …