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

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In this paper a generalized inner product 〈x|y〉 is defined as a binary function with complex values which satisfies the following: (i) for any nonzero vector y and any complex number ζ there exists a vector x such that 〈x|y| = ζ, (ii) 〈x1 + x2|y1 + y2〉 = 〈x1|y1〉 + 〈x2|y1〉 + 〈x1|y2〉 + 〈x2|y2〉, (iii) 〈y|x〉 = f[〈x|y〉], where f is a continuous function and, (iv) 〈x|μy〉 = g[μ, 〈x|y〉], where g is a continuous function. These conditions induce several functional equations which are then solved. By making a linear combination of 〈x|y〉 with its complex conjugate a new function (x|y) is obtained which is either symmetric, antisymmetric, or Hermitian. The functions 〈x|y〉 and (x|y) have the same orthogonal vectors. © 1973.

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Linear Algebra and Its Applications