In mathematics, more specifically in functional analysis, a Banach space 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 within the space.
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a convex compact subset of Euclidean space to itself.
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
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 hyperplane separation theorem, and has numerous uses in convex geometry.
In mathematics, an extreme point of a convex set in a real or complex vector space is a point in which does not lie in any open line segment joining two points of In linear programming problems, an extreme point is also called vertex or corner point of
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations.
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if is a nonempty convex closed subset of a Hausdorff topological vector space and is a continuous mapping of into itself such that is contained in a compact subset of , then has a fixed point.
Czesław Ryll-Nardzewski was a Polish mathematician.
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs).
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint.
In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.
In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of abelian groups.
In mathematics, Kostant's convexity theorem, introduced by Bertram Kostant (1973), states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of Schur (1923), Horn (1954) and Thompson (1972) for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all n by n complex self-adjoint matrices with given eigenvalues Λ = is the convex polytope with vertices all permutations of the coordinates of Λ.
In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.
Isaac Namioka was a Japanese-American mathematician who worked in general topology and functional analysis. He was a professor emeritus of mathematics at the University of Washington. He died at home in Seattle on September 25, 2019.
In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski.
In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide. For a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim, and, soon after, under the name of almost convergence, by Tadeusz Kuczumow.
The Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for uniformly convex Banach spaces. It asserts that if is a nonempty convex closed bounded set in uniformly convex Banach space and is a mapping of into itself such that , then has a fixed point.
