ABSTRACT
In this chapter, the authors introduce one way of forming the product of vectors, called the inner product, that associates a scalar with each pair of vectors. Among other things, an inner product enables us to generalize the notion of length that occurs in Euclidean spaces. The terms “inner product” and “dot product” are often used synonymously, but they are not exactly the same. Jurgen Gram mathematical career was a balance between pure mathematics and practical applications of the subject. The practical applications to forestry, which he continued to study at this time, and his work on probability and numerical analysis involved both the theory and its application to the practical situations. For inner product spaces finding the representation of a linear transformation with respect to given bases is easier if the basis of the image vector space is orthonormal.
