It is not clear to me what link there is between the "middle third" Cantor space and a Hilbert space. Here's what I have:
(a) every compact metric space is a continuous image of the Cantor set
(b) the spectrum of an operator in Hilbert space is a compact set
Seems a rather tenuous connection. Anything better?
Secondly, (a) above seems to be proven essentially by
f: (Cantor set)-->2^omega<--> I=[0,1]
Hilbert space is a subset of I^n
But that seems rather weak, because I can take lots of sets and make a function onto I. What is so special about the Cantor set? It is nowhere dense -- how does this help with Hilbert space or the space of continuous functions in Hilbert space?

As you can see, I am confused about the connection. Any clarity in the fog would be appreciated.