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Zbl 0554.47001
Brunner, Norbert
Hilberträume mit amorphen Basen. (German)
[J] Compos. Math. 52, 381-387 (1984). ISSN 0010-437X; ISSN 1570-5846/e

Ramsey's theorem RT is the assertion, that for every infinite set X, every set $C\subseteq [X]\sp 2$ (the family of pairs) there is an infinite $H\subseteq X$ such that $[H]\sp 2\subseteq C$ or $[H]\sp 2\cap C=\emptyset.$ RT is a weak form of the axiom of choice which is consistent with the existence of an amorphous set (an infinite set each of its infinite subsets is cofinite). RT implies, that every operator on a Hilbert space with an amorphous base is a direct sum of a finite matrix and a scalar operator. RT is necessary to obtain this conclusion.

MSC 2000:: *47A05 General theory of linear operators
03E25 Axiom of choice and related propositions (logic)
46C05 Geometry and topology of inner product spaces

Keywords: Ramsey's theorem; weak form of the axiom of choice; existence of an amorphous set; every operator on a Hilbert space with an amorphous base is a direct sum of a finite matrix and a scalar operator

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