As teaching material for this course we will primarily use the book "Topics in Real and Functional Analysis" by Gerald Teschl. This book is available online. You can download a pdf-version of it here.
In addition, we will cover some Fourier analysis. A short set of lecture notes will be available online.
Moreover, the exercises in Teschl's book will be supplemented by additional exercises. These will be announced on the course homepage, under the detailed teaching plan.
We will cover large parts of chapters 1, 2, 3, 5, 7 and 8 in Teschl's book.
The following part of the theory is relevant for exam.
Teschl's book:
Chp.1 section 1.1 except 1.7, 1.8, 1.16, 1.17; section 1.2 except Schauder basis and proof of 1.20; section 1.3 except proof of converse of 1.23; section 1.5 except 1.31.
Chp. 2 section 2.1 except 2.7, section 2.2, section 2.3 except 2.14 and 2.15; section 2.4 (only up to the first example).
Chp. 3 section 3.1 except 3.3 and proof of 3.5; section 3.2, section 3.3.
Chp. 5 section 5.1 up to (and including) Corollary 5.3; section 5.2 Lemma 5.5 and compactness of a Hilbert-Schmidt operator (for definition see the Report from lectures); section 5.3 up to (and including) 5.11.
Chp. 7 section 7.1 except general distribution functions and supports of measures; section 7.2; section 7.3 except lower and upper semicontinuous functions; section 7.4 inner regularity, Corollary 7.15, for Lebesgue measure on the Borel sigma-algebra on R; section 7.5; section 7.7 only 7.31 and the example with an affine transformation on R.
Chp. 8 section 8.1; section 8.2 except 8.2 and 8.3; section 8.3 except 8.8 and 8.12, and with X=R and \mu= Lebesgue measure on the Borel sigma-algebra in theorems 8.9 and 8.11.
All of the material in the "Lecures Notes on Fourier Series".