Vol. 82, No. 1, 1979

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ISSN: 0030-8730
Subspaces of positive definite inner product spaces of countable dimension

Werner Bäni

Vol. 82 (1979), No. 1, 1–14
Abstract

We deal with the following problem, proposed by I. Kaplansky in 1950: If V , V are subspaces of an inner product space (E,Φ) of countable dimension over any field k, when does there exist a metric automorphism of (E,Φ) mapping V onto V ? The present paper treats the case of positive definite symmetric spaces over k = R. We shall characterize the orbits (under the orthogonal group of (E,Φ)) of a large class of subspaces V by two sequences of cardinals attached to V in a natural way (if e.g., V = 0 or V = V ⊥⊥ only a few of them are 0; the case V = 0 is covered by work of H. Gross). However, classifying the subspaces not in this class is equivalent to classifying vector spaces F endowed with a sequence Ω0,Ω1,Ω2, of positive definite forms.

Mathematical Subject Classification 2000
Primary: 10C04, 10C04
Secondary: 15A63, 46C99
Milestones
Received: 18 September 1978
Published: 1 May 1979
Authors
Werner Bäni