ABSTRACT
In the section on Euclidean spaces we defined a scalar product on a vector space V over 𝕂 = ℝ or ℂ as a mapping V × V → 𝕂 with certain properties. Now many results we obtained for scalar products (like the Cauchy-Schwarz inequality, orthogonal decompositions, the Riesz representation theorem, and so on) can be seen to only depend on some, but not on all the properties of a scalar product. The attempt to extend the notion of a scalar product to a more general situation leads to the following definitions.
