# Hilbert Space as direct sum of subspaces with cyclic vectors

Ok,so this should be easy, however I havent taken functional analysis for a while. But given a compact self-adjoint operator on a hilbert space H(over the complex numbers), we define v to be a cyclic vector if and only if the family q(A)v for complex polynomials is dense in H. Halmos, in the article "What does the spectral theorem say?", claims that a hilbert space (additional assumptions on its structure?) can be decomposed as a direct sum of subspaces so that the restriction of A to these spaces has a cyclic vector. He says it can be proved by "a standard transfinite argument." Well Im not up on my standards, and I was wondering if someone could break this problem down (pun intended) for me, preferably as suggesting the tools Ill need to carry out a proof?

-