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.
