Vol. 82, No. 1, 1979

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Subspaces of positive definite inner product spaces of countable dimension

Werner Bäni

Vol. 82 (1979), No. 1, 1–14

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
Received: 18 September 1978
Published: 1 May 1979
Werner Bäni