# The Hahn-Hellinger Theorem [closed]

People can tell the question is not up to the mark, or research level question, but I felt without understanding the Hahn-Hellinger Theorem properly there is no point talking von Neumann algebras for myself. So can any body help me out clearing the concept of Hahn-Hellinger Theorem, what is the game playing inside the theorem, I tried many books and I am not able to get with clarity. More specifically how cyclic vectors are connected with multiplicity of self-adjoint operator, this theorem say any self-adjoint operator in $\mathcal{H}$ is equivalent to a multiplication operator $M_{z}$ on $\oplus L^2(X,\mu_{i})$, how it looks the case at least for finite dimensional case, how direct sum take cares multiplicity of the eigenvalue that I did not get, Also it is not clear all the measure joined to single measure. Please help. Thanks in advance

## closed as too broad by Francois Ziegler, Yemon Choi, Jan-Christoph Schlage-Puchta, Pace Nielsen, András BátkaiAug 6 '18 at 17:56

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• In my view, the problem with your question is not so much "it is not research level" but that it is far too vague. MO is not an online tutoring service nor a place to get people to write Wikipedia entries. Instead of saying "I don't understand this theorem, please explain the concept", a better question on MO would be to find some specific aspect of the proof which you don't understand, or to ask about some specific analogy which you have in mind – Yemon Choi Jul 20 '18 at 11:00
• Okk, I am trying to be more specific, so the thing is I am reading the hahn-hellinger Theorem from V.S.Sunder's book. I don't understand how cyclic vectors are connected with multiplicity of self-adjoint operator, this theorem say any self adjoint operator in $\mathcal{H}$ is equivalent to a multiplication operator $M_{z}$ on $\oplus L^2(X,\mu_{i})$, how it looks the case atleast for finite dimensional case, how direct sum take cares multiplicity of the eigenvalue that I did not get, Also it is not clear all the measure joined to single measure – mathlover Jul 20 '18 at 11:13
• Thanks: I suggest you edit this extra information into your original question – Yemon Choi Jul 20 '18 at 11:18
• People can see the theorem given in Arveson "An invitation to $C*$-algebras" on the multiplicity theory chapter, I got complaints many people have not heard this – mathlover Jul 29 '18 at 15:29