In this paper: 'Borel structures for Function spaces' by Robert Aumann,

http://projecteuclid.org/euclid.ijm/1255631584

Aumann claims that when X and Y are metric spaces (among other things), the Baire classes and Banach classes coincide, I've been looking all over and can't find a reference or proof, I need to know if the result can be extended to METRIZABLE spaces (that is, if the Baire classes and Banach classes coincide for metrizable spaces too), I desperately need this to be true for a proof I'm working on, it's like the last step in my proof, so I was wondering if anyone knows this or can point me in the right direction? In Aumann's paper There's a reference to this paper: 'Über analytisch darstellbare Operationen in abstrakten Räumen' by Banach, but it's in german and I don't speak it and as far as I know there's no translation to english.

Classical Descriptive Set Theory, Theorem 24.3. This proof makes essential use of separability of $Y$. – Theo Buehler Jun 23 '11 at 6:20