Chapter 0

A first look at Banach

and Hilbert spaces

I assume that the reader has some basic familiarity with measure theory and func-

tional analysis. For convenience, some facts needed from Banach and

Lp

spaces

are reviewed in this chapter. A crash course in measure theory can be found in

Appendix A. If you feel comfortable with terms like Lebesgue Lp spaces, Banach

space, or bounded linear operator, you can skip this entire chapter. However, you

might want to at least browse through it to refresh your memory.

0.1. Warm up: Metric and topological spaces

Before we begin, I want to recall some basic facts from metric and topological

spaces. I presume that you are familiar with these topics from your calculus

course. As a general reference I can warmly recommend Kelly’s classical

book [33].

A metric space is a space X together with a distance function d :

X × X → R such that

(i) d(x, y) ≥ 0,

(ii) d(x, y) = 0 if and only if x = y,

(iii) d(x, y) = d(y, x),

(iv) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).

If (ii) does not hold, d is called a pseudometric. Moreover, it is straight-

forward to see the inverse triangle inequality (Problem 0.1)

(0.1) |d(x, y) − d(z, y)| ≤ d(x, z).

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