A book for problems in Functional Analysis

Another classical book is Theorems and problems in functional analysis by Kirillov and Gvishiani.

MR0675952 (84e:47001) Halmos, Paul Richard A Hilbert space problem book. Second edition. Graduate Texts in Mathematics, 19. Encyclopedia of Mathematics and its Applications, 17. Springer-Verlag, New York-Berlin, 1982. xvii+369 pp. ISBN: 0-387-90685-1

Also Brezis' book Functional analysis, Sobolev spaces and partial differential equations contains a lot of nice and interesting functional analytical problems.