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Krein-Smulian Theorem: [2] — Let be a Banach space and a weakly compact subset of (that is, is compact when is endowed with the weak topology). Then the closed convex hull of K {\displaystyle K} in X {\displaystyle X} is weakly compact.
For this reason, the following corollary to the above theorem is also often called the Krein–Milman theorem. ( KM ) Krein–Milman theorem (Existence) [ 2 ] — Every non-empty compact convex subset of a Hausdorff locally convex topological vector space has an extreme point ; that is, the set of its extreme points is not empty.
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality , between compact and discrete commutative topological groups, to groups that are compact but noncommutative .
to be dense in a weighted L 2 space on the real line. It was discovered by Mark Krein in the 1940s. [1] A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem. [2] [3]
Let be a Banach space, and let be a convex cone such that = {}, and is dense in , i.e. the closure of the set {:,} =. is also known as a total cone.Let : be a non-zero compact operator, and assume that it is positive, meaning that (), and that its spectral radius is strictly positive.
Mark Grigorievich Krein (Ukrainian: Марко́ Григо́рович Крейн, Russian: Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
If is a reflexive Banach space then this conclusion is also true when = [2]. Metric reformulation. As usual, let (,):= ‖ ‖ denote the canonical metric induced by the norm, call the set {: ‖ ‖ =} of all vectors that are a distance of from the origin the unit sphere, and denote the distance from a point to the set by (,) := (,) = ‖ ‖.