This course consists of two parts:
Part I: Measure and integration
Part II: Bounded linear operators on Banach spaces and Hilbert spaces.
Literature for Part I: the book "A course in real analysis" by John McDonald and Neil Weiss, 2nd edition. Final syllabus consists of the following sections: 1.4 (up to Def 1.16), 3.1, 3.2, 3.4, 4.1, 4.2, 4.3 (we saw a slightly different proof of Theorem 4.9 that did not use Lemma 4.4), 4.4, 5.1 (except Thm 5.2), 5.2, 5.3, 5.4 (up to Corollary 5.3), product measure (see the lecture notes or section 6.3), 6.4 (theorems of Tonelli and Fubini, see the notes for a different proof), 13.4 (up to and including Thm 13.11). Section 14.3 (which overlaps with parts of chapters 6 and 7 in Rynne and Youngson).
Literature for part II: the book "Linear Functional Analysis" (2nd edition) by Bryan P. Rynne and Martin A. Youngson, Springer Undergraduate Mathematics Series, ISBN 978-1-84800-004-9. Syllabus: 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 3.4, 3.5, 4.1 (except 4.12 and 4.13), 4.2, 4.3, 5.1 (only Thm 5.2), 6.1 (up to Corollary 6.12), 6.2 (starting with Def. 6.21), 6.4 (up to and including Corollary 6.53), 7.1 (up to Thm 7.12), 7.2 (only Def. 7.17, Thm 7.19, Thm 7.22, Corollary 7.23), 7.3 (up to and including Corollary 7.36).
A good part of the material on measure and integration theorys is also covered in chapters 5 and 6 of Tom Lindstr?m's notes for MAT2400 (spring 2013). These chapters can be downloaded here .