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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.
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 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.
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 ]
The Hahn–Banach theorem asserts that φ can be extended to a linear functional on V that is dominated by N. To derive this from the M. Riesz extension theorem, define a convex cone K ⊂ R×V by = {(,) ()}. Define a functional φ 1 on R×U by
Theorem. The Gelfand–Naimark representation of a C*-algebra is an isometric *-representation. The Gelfand–Naimark representation of a C*-algebra is an isometric *-representation. It suffices to show the map π is injective , since for *-morphisms of C*-algebras injective implies isometric.
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 ...