Vector Spaces and Inner Products

Abstract

Clifford algebras and spinor spaces occurring in physics are vector spaces over the real or complex numbers. Quaternions also appear, in a natural manner, as was alluded to in the Introduction. There are subtle relations between these number fields and the signature of the quadratic form under consideration. In some cases, there is a “charge conjugation” which allows the definition of real spinors. To prepare ground for a systematic presentation of such matters, we summarize here some elementary notions related to the introduction of real, complex and quaternionic structures in vector spaces. We also review the definitions and basic properties of inner products and Hermitean forms needed in the sequel.