On the types of functions which can serve as scalar products in a complex linear space

Abstract

In this paper a generalized inner product (xly) is defined as a binary function with complex values which satisfies the following: (i) for any nonzero vector y and any complex number 5 there exists a vector x such that (x/y) = c, (ii) (xi + xslyr + ys) = (x,ly,) + (~,IY,) + (4~~) + (x,ly,). (4 (ylx) = f[(xly)l, where f is a continuous function and, (iv) (+Y) = g[p, (4y)l, where g is a continuous function. These conditions induce several functional equations which are then solved. By making a linear combination of (xly) with its complex conjugate a new function (xly) is obtained which is either symmetric, antisymmetric, or Hermitian. The functions (xly) and (xly) have the same orthogonal vectors.