TMA4230 Functional analysis 2006
Syllabus (Pensumliste)
Basically, the syllabus consists of Chapters 4, 7 and 9 of Erwin Kreyszig: Introductory functional analysis with applications plus my own Assorted notes on functional analysis (20060511 version).
However, there are the following exceptions and amendments to the above list:

My notes, Ch 1 (transfinite induction)

All you are expected to know is the statement of Zorn's lemma (and the definitions needed to understand it), plus some basic notion on how it is used.

My notes, Ch 4 (topology)

No direct questions will be asked, but you will need a basic understanding of topological concepts like open and closed sets, continuity and compactness. You should know the statement of Tychonov's theorem and the Weierstrass density theorem. (We haven't looked at normal spaces and Urysohn's lemma at all.)

My notes, Ch 5 (topological vector spaces)

We had no time for the Krein–Milman theorem or Milman's converse. (A pity, for these are beautiful and useful results.)

My notes, Ch 7 (compact operators)

This chapter is not included.

Kreyszig section 4.4 (bounded linear functionals on C[a,b])

No need to understand this example in detail.

Kreyszig section 4.7 (convergence of sequences of operators and functionals)

We have barely touched on this, so I won't ask about it. But for future work in functional analysis, these definitions are important.

Kreyszig section 4.10 (summability of sequences)

We skipped it.

Kreyszig section 7.2 (basic concepts of spectral theory)

Forget all about operators that aren't defined on the entire space X. (This is really important only when working with unbounded operators, which is an important topic we have not visited.) Likewise, don't worry about point spectrum, continuous spectrum, and residual spectrum. But you should know that not all points of the spectrum are eigenvalues.

Kreyszig's 7.42 (spectral mapping theorem for polynomials)

This proof is hopelessly overcomplicated. See the proof of Proposition 74 in my notes instead.

Kreyszig section 9.3 (positive operators)

Don't bother with the part on monotone sequences. In any case, I presented a shorter proof of theorem 9.33 using the polarization identity. I haven't written it up, so I won't ask.

Kreyszig section 9.4 (square roots of a positive operator)

Skip this section. In my notes, the full spectral theory is developed, which makes the existence of the square root trivial.
You may also note that my notes and Kreyszig's book overlap in a great many places. The most important ones are these:

Uniform boundedness theorem

The proof in Ch 2 of my notes does not use the Baire category theorem, unlike Kreyszig's proof in 4.73.

Spectral theory

Chapter 6 of my notes started out as a supplement to Kreyszig, but now this chapter stands pretty much on its own, and replaces large portions of Kreyszig's Chapters 7 and 9. I suggest using my notes as the primary source and to consult Kreyszig's chapter 7 and 9 for a different perspective and additional details where necessary.