Syllabus :
The relevant material for the exam consists of the following chapters from the book Linear Functional Analysis, by Rynne and Youngson:
Chapter 1, chapter 2, chapter 3, chapter 4 (except the proofs of 4.43, 4.44, 4.45, 4.52 and 4.53), section 5.1, section 5.3 (only 5.19, 5.21 and 5.22. The proof of 5.19 is not included), section 5.5 (only up to 5.41), chapter 6, chapter 7, and, in addition to this, the exercises that have been assigned during the semester. Also, look at all other exercises in the book which are relevant to the theory as listed above.
Suggestions for additional reading:
Peter D. Lax: Functional Analysis, 2002. Wiley. ISBN: 0-471-55604-1. Lax's book covers the syllabus requirements for the courses MAT4340 and MAT4350, and goes deeper into applications. However, it contains a lot more material than required for these courses. Nevertheless, it is recommended as additional reading for all those who wish to study further topics in analysis.
John Conway, An introduction to Functional Analysis, Springer-Verlag. (Covers more material than the book by Rynne and Youngson, but is still pedestrian.)
Gert K. Pedersen, Analysis Now, Springer-Verlag. (This book covers material on topology, normed spaces and duality, Hilbert spaces, operator algebras, unbounded operators, and a chapter on measure and integration. The style is elegant but condensed, and the material is particularly well organised.)