A Fresh Look at Vector Spaces

We start at the level of vector spaces, and we first consider quite generally a vector space as it is given only by its definition, an abstract vector space. The most appropriate way is to use maps that are in harmony, that is, compatible with a vector space structure. We have to use linear maps, also called vector space homomorphisms. It turns out that the physical reality demands additional structures which we have to impose on an abstract vector space. The most prominent structure of this kind is a positive definite scalar product (special symmetric bilinear form), the inner product, and in this way we obtain an inner product vector space or a Euclidean vector space which is strongly connected with our well-known (affine) Euclidean space. We have, of course, also semi-Euclidean vector spaces where the scalar product is no more positive definite. It is interesting that instead of adding, as in Sect. 2.3 with a symmetric bilinear form, we could also “subtract” structures from vector spaces. This means we can consider a vector space a “special” manifold, a linear manifold which is usually called affine space.