Operators on Inner Product Spaces

In this chapter, we summarize and complete some important facts about inner product spaces, real and complex ones, mostly on a more advanced level than in Chaps. 2 and 6 . In this context, the notions of orthogonality, orthogonal compliment, orthogonal projection, and orthogonal expansion are discussed once again. In Sect. 10.4, we motivate and discuss normal operators in some detail. We give both an algebraic and a geometric definition of normal operators. We explain a surprising and gratifying analogy of normal operators to complex and real numbers. The highlight of this chapter and in fact of linear algebra altogether, are the spectral theorems that are treated at the end of this chapter.