Jordan identity, von Neumann identity, functional equations, characterization of Hilbert space, bilinear function, Inner products, Functional equations for real functions, Ptolemaic inequality, Functional equations for functions with more general domains and/or ranges, Equacions funcionals, von Neumann characterization, inner product space, inner product spaces, Jordan characterization, Fréchet characterizations, Classificació AMS::39 Difference and functional equations::39B Functional equations and inequalities, Functional equations, characterizations, biadditive function, additive function, normed linear space, Characterizations of Hilbert spaces, Fréchet condition, Quadratic and bilinear forms, inner products