ABSTRACT
We have seen how all the metric notions of geometry can be defined in a Euclidean vector space using the scalar product. In a field K different from R, the property of being positive definite has no meaning, and so it is not possible to define a useful scalar product for general fields. However, in the case K = C it is possible to modify the definition of symmetric bilinear form to that of ‘Hermitian form’, and we will see that many of the properties of Euclidean spaces can be extended, using positive definite Hermitian forms, to ‘Hermitian spaces’.
