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). 3
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