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In mathematics, particularly in functional analysis, the Krein-Smulian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan , who published them in 1940.
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 ...
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 .
In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums {= (),,},to be dense in a weighted L 2 space on the real line.
Theorem — The set of states of a -algebra with a unit element is a compact convex set under the weak-topology. In general, (regardless of whether or not A {\displaystyle A} has a unit element) the set of positive functionals of norm ≤ 1 {\displaystyle \leq 1} is a compact convex set.
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The Krein–Milman theorem is arguably one of the most well-known theorems about extreme points. Krein–Milman theorem — If S {\displaystyle S} is convex and compact in a locally convex topological vector space , then S {\displaystyle S} is the closed convex hull of its extreme points: In particular, such a set has extreme points.
Krein space Krein's condition Krein's extension theorem Krein–Milman theorem Krein–Rutman theorem Krein–Smulian theorem Akhiezer–Krein–Favard constant Markov–Krein theorem Tannaka–Krein duality: Awards: Wolf Prize (1982) Scientific career: Fields: Operator theory Mathematical Physics: Institutions: I.I. Mechnikov Odesa National ...