In this lecture and the next few lectures we will study the basic theory of Hilbert spaces. Hilbert spaces are usually studied over or over . In this course, whenever we consider Hilbert spaces, we shall consider only complex Hilbert spaces, that is, spaces over . The are two reasons for this. First, the results in this post hold equally well for real Hilbert spaces with similar proofs. Second, in some topics that we will discuss later the nice results only hold for complex spaces. So we will ignore real Hilbert spaces because they are essentially the same and also because they are fundamentally different!
Remark: The only situation I know where it is really important to concentrate on real Hilbert spaces when doing convex analysis (there must be others that I don’t know of). On the other hand, it is often convenient – indeed, we already did so in this course – to study real Banach spaces.