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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.
Krein–Milman theorem [2] — Suppose is a Hausdorff locally convex topological vector space (for example, a normed space) and is a compact and convex subset of . Then K {\displaystyle K} is equal to the closed convex hull of its extreme points : K = co ¯ ( extreme ( K ) ) . {\displaystyle K~=~{\overline {\operatorname {co ...
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]
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 .
The finite-dimensional Krein–Milman theorem, which is due to Minkowski, can be quickly proved using the concept of -extreme points. If S {\displaystyle S} is closed, bounded, and n {\displaystyle n} -dimensional, and if p {\displaystyle p} is a point in S , {\displaystyle S,} then p {\displaystyle p} is k {\displaystyle k} -extreme for some k ...
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 Krein–Rutman theorem is a generalisation of the Perron–Frobenius theorem to infinite-dimensional Banach spaces. [1] It was proved by Krein and Rutman in 1948. [ 2 ]
Let be a real vector space, be a vector subspace, and be a convex cone.. A linear functional: is called -positive, if it takes only non-negative values on the cone : ().A linear functional : is called a -positive extension of , if it is identical to in the domain of , and also returns a value of at least 0 for all points in the cone :