ABSTRACT
The starting point for the study of vector spaces came from geometry, the prototype of a vector space being the real vector space V of all equivalence classes of arrows. We saw in sections 1 and 12 how geometric problems could be solved by using vectors; whereas we studied only affine properties in section 1, we concentrated on metrical properties in section 12. Now it will turn out that many of the metrical concepts we encountered in section 12 can be generalized to vector spaces other than V, provided that the base-field is β or β. This is useful because it makes our geometric intuition applicable to the study of vector spaces other than V. The key concept we want to introduce is that of a general scalar product. Throughout this section we write π for a base-field which is either β or β.
