On Inner Products in Linear, Metric Spaces

Abstract

1. In his foregoing paper' Mr. M. Frechet discussed the following question: When is a linear, metric space L isometric with a generalized Hilbert space?2 In other words: When can one define in it a bilinear symmetric inner product, from which its given metric can be derived in the customary way? Mr. Frechet discovered a necessary and sufficient algebraic condition, from which he derived this more abstract criterium: This is the case if and only if every < 3-dimensional linear subspace L' of L is isometric with a Euclidean space. On the pages which follow we will derive another necessary and sufficient algebraic condition, which implies that Mr. Fr6chet's abstract criterium can be weakened as follows: The answer to the above question is affirmative if and only if every ? 2-dimensional linear subspace L' of L is isometric with a Euclid-ean space. The criterium which we will derixve has some further interest, because it shows that the linear spaces with a bilinear symmetric inner product are in a certain sense limiting cases of the general linear metric spaces. We will only consider complex linear spaces. Real linear spaces could be discussed along the same lines, even with some simplifications. Mr. Frdchet showed, loc. cit., how the two types of linearity are connected. 2. We repeat the customary definitions of linearity, metricity, and of the bilinear symmetric inner products. As we consider complex linearity, the symmetry of inner products will be interpreted as Hermitian symmetry. The definitions in question are: DEFINITION 1.3 A space L is (complex) linear and metric, if for all f, g e L an 1 Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicable vectoriellement sur l'espace de Hilbert, Ann. of Math. (2) 36 (1935) pp. 705-718. 2 Hilbert space is uniquely characterized (up to an isomorphism) by five postulates A-E, which have been formulated by one of us, Math. Ann. vol. 102 (1929), pp. 63-66. Of these postulates A, B are the really essential ones: The omission of C would only add the (finite dimensional) Euclidean spaces; the omission of D could be compensated by the standard manipulation of "completion;" in the absence of E essentially new hyper-Hilbert spaces arise, but they are nevertheless similar to Hilbert space under most aspects. The hyper-Hilbert spaces (without E) have been first discussed by H. L5wig, Acta Szeged, vol. 7 (1934), pp. 1-33. The "completion" of spaces without D plays a fundamental role in the work of K.