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Tannaka's theorem then says that this map is an isomorphism. Krein's theorem answers the following question: which categories can arise as a dual object to a compact group? Let Π be a category of finite-dimensional vector spaces, endowed with operations of tensor product and involution.
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.
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, 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.
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 ]
Kirchberger's theorem (discrete geometry) Krein–Milman theorem (mathematical analysis, discrete geometry) Minkowski's theorem (geometry of numbers) Minkowski's second theorem (geometry of numbers) Minkowski–Hlawka theorem (geometry of numbers) Monsky's theorem (discrete geometry) Pick's theorem ; Pizza theorem ; Radon's theorem (convex sets)
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.
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.