Semi-inner-product spaces

Abstract

function as a particular Banach space (whose norm satisfies the parallelogram law), but rather as an inner-product space. It is in terms of the innerproduct space structure that most of the terminology and techniques are developed. On the other hand, this type of Hilbert space considerations find no real parallel in the general Banach space setting. Some time ago, while trying to carry over a Hilbert space argument to a general Banach space situation, we were led to use a suitable mapping from a Banach space into its dual in order to make up for the lack of an innerproduct. Our procedure suggested the existence of a general theory which it seemed should be useful in the study of operator (normed) algebras by providing better insight on known facts, a more adequate language to classify special types of operators, as well as new techniques. These ideas evolved into a theory of semi-inner-product spaces which is presented in this paper (together with certain applications)(1). We shall consider vector spaces on which instead of a bilinear form there is defined a form [x, y] which is linear in one component only, strictly positive, and satisfies a Schwarz inequality. Such a form induces a norm, by setting |x| = ([x, x])112; and for every normed space one can construct at least one such form (and, in general, infinitely many) consistent with the