Euclidean and Unitary Vector Spaces

Abstract

In this chapter we study vector spaces over the fields $\mathbb R$ and $\mathbb C$. Using the definition of bilinear and sesquilinear forms, we introduce scalar products on such vector spaces. Scalar products allow the extension of well-known concepts from elementary geometry, such as length and angles, to abstract real and complex vector spaces. This, in particular, leads to the idea of orthogonality and to orthonormal bases of vector spaces. As an example for the importance of these concepts in many applications we study least-squares approximations.