Inner Product Spaces

We have seen in Chapter VI that we can build, by an algebraic structure a linear vector space out of a set, especially out of a set of functions, and we can further endow this space with a topological structure generated by a real-valued norm function that may be regarded as measuring the length of each vector. Thus, we can start to employ a geometrical language. Then it becomes possible to identify with comparative ease rather general features of such spaces that could prove to be quite effective in extracting very useful concrete results in various applications. If we wish to continue thinking in terms of geometrical concepts, one naturally expects that we should attempt to define some kind of an angle between the elements of a vector space. In three-dimensional vector spaces, which are frequently encountered in physical and engineering applications, the angles between vectors are measured by the scalar product. The scalar product of two vectors was defined by the American mathematical physicist Josiah Willard Gibbs (1839–1903), who also defined the vectorial product, and also independently, by the English engineer Oliver Heaviside (1850–1925) in the early 1880’s. They were in turn inspired by the quaternion product introduced by Hamilton and by the algebra proposed by the German mathematician Hermann Günter Grassmann (1809–1877). It then became a rather easy task to generalise this concept to any finite-dimensional real vector space.