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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 is equal to the closed convex hull of its extreme points: = ¯ ( ()).
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's theorem on non-negative self-adjoint extensions [ edit ] M. G. Krein has given an elegant characterization of all non-negative self-adjoint extensions of a non-negative symmetric operator T .
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.
A corollary, also called Krein's condition, ... This can be derived from the "only if" part of Krein's theorem above. [4] Example. Let = ...
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 ...
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 ...
Tannaka–Krein duality concerns the interaction of a compact topological group and its category of linear representations. Tannaka's theorem describes the converse passage from the category of finite-dimensional representations of a group G back to the group G, allowing one to recover the group from its category of representations. Krein's ...