Cantor set and Hilbert space?

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. (Headbang) Any clarity in the fog would be appreciated.