Are there some good questions on functional analysis whose solution depends on tools in set theory? My major is mathematical logic, I think tools in set theory, especially infinity combinatorics and forcing, should be used to solve some questions in functional analysis. For functional analysis, I just have read the main part of Conway's textbook. In this book, I have not found such questions.

One of the most elementary examples is the use of the infinite version of Ramsey's theorem to prove Rosenthals $\ell_1$ theorem. See Chapter 10 in Albiac & Kalton, Topics in Banach Space Theory. You'll find ultraproducts in Chapter 11. More model theory than set theory, but still logic. Deeper set theory (large cardinals, for example) have also been used. Check out, e.g., Todorcevic, S* on MathSci Net for recent things. 


The survey article ``Set theory and C*algebras'' by Nik Weaver might have some things along the lines you are looking for ( Bull. Symbolic Logic Volume 13, Issue 1 (2007), 120; see also math/0604198 on the arXiv). For a particular example of Weaver's recent work in this area, see Akemann & Weaver's paper Consistency of a counterexample to Naimark's problem We construct a C*algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assuming those axioms are consistent). We prove that the statement ``there exists a counterexample to Naimark's problem which is generated by $\aleph_1$ elements'' is undecidable in standard set theory. There has also been some work by I. Farah on applying settheoretical technqiues to operatoralgebraic problems. 


One interesting example is A discontinuous homomorphism from C(X) without CH, by W. Hugh Woodin, which begins with the following introduction.



Although the construction of Tsirelson's space doesn't use set theory per se, in this short essay Tsirelson recounts (among other things) how his construction was inspired by forcing. 


Chris Phillips and Nik Weaver wrote a paper called The Calkin Algebra has Outer Automorphisms, where they showed that the Continuum Hypothesis implies that the Calkin algebra $\mathcal{B(H)/K(H)}$ has outer automorphisms. See also this paper of Farah, McKenney, and Schimmerling, and references therein; they show that it is relatively consistent with ZFC that the Calkin algebra has only inner automorphisms, and hence the question of existence of outer automorphisms is independent of ZFC. 


Though this is a little more advanced, there is actually some very exciting research right now at the intersection of descriptive set theory, ergodic theory, and von Neumann algebras. It is quite striking that the three areas have powerful tools for looking at similar problems, and yet tend to be applicable in different cases. For a nice introduction to some of these ideas from a more settheoretical point of view I would say check out "Topics in Orbit Equivalence" by Kechris and Miller. http://www.springerlink.com/content/0pwfmbrandag/ Here is a link where you can download it (though you might need a subscription but many universities will have it so it should work on a department computer.) It is actually quite elementary, you need some basic descriptive set theory and measure theory, but arrives at quite deep theorems. 

