# For METRIZABLE spaces, do the Banach classes and Baire classes coincide?

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.

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The best thing you can do that I'm aware of is the case where $X$ and $Y$ are metrizable and $Y$ is in addition separable. A proof can be found in Section 24 of Kechris, Classical Descriptive Set Theory, Theorem 24.3. This proof makes essential use of separability of $Y$. – Theo Buehler Jun 23 '11 at 6:20
Thank you Theo, I finally figured it out, cheers – Mario Carrasco Jul 26 '11 at 14:04
And, since there is no answer, the system will keep re-posting to the front page from time to time (as it did today). – Gerald Edgar Feb 25 '14 at 17:38