Lawrence C. Evans: Partial Differential Equations, 1998. American Mathematical Society. ISBN:?0-8218-0772-2 (alk. paper). Most of the content of Chapter 5, some of Chapter 6.
The appendices in Evans cover well the material from real analysis, functional analysis and measure theory which is needed for the course. These appendices are A.1. A.2, A.3, A.4, A.5, A.6, B.2 c e f h i, C.2, C.3 (just Theorem 4), C.4, C.7, D.1, D.2, D.3 a (just the part on linear functional analysis), D.3 b (just Riesz representasjon's theorem), E.1, E.2. E.3.
Kendall Atkinson and Weimin Han: Theoretical numerical analysis, 2009. Springer. ISBN:?978-1-4419-0457-7. Contents from Chapter 7, 9 and 10. Available online.