Chapter 2 - metric vector spaces

Abstract

This chapter focuses on metric vector spaces. Affine geometry is the study of incidence and parallelism. A vector space, provided with an inner product, is called a metric vector space, a vector space with metric or even a geometry. It is very important to adopt the geometric attitude toward metric vector spaces. This is done by taking the pictures and language from Euclidean geometry. Two vectors A and B of a metric vector space are orthogonal—perpendicular—if AB = 0. The definition does not exclude the possibility that one or both of the vectors A, B is equal to the origin 0. In Euclidean geometry, the only vector orthogonal to all vectors is the origin. A metric vector space is called nonsingular if the origin is the only vector which is orthogonal to all vectors. In general, two mathematical structures are called “equivalent” or isomorphic if there exists a one-to-one mapping from one of the structures onto the other one which preserves all intrinsic structure. Such a mapping is called an isomorphism and hence, two structures are isomorphic if there exists an isomorphism between them.