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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.
Download as PDF; Printable version; ... In mathematical analysis, Krein's condition provides a ... This can be derived from the "only if" part of Krein's theorem ...
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 ...
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.
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 original Krein–Milman theorem follows from Choquet's result. Another corollary is the Riesz representation theorem for states on the continuous functions on a metrizable compact Hausdorff space. More generally, for V a locally convex topological vector space, the Choquet–Bishop–de Leeuw theorem [1] gives the same formal statement.
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 ]
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 ...