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Definition: Polish space is a separable completely metrizable topological space. Does it mean that a Polish space can be infinite dimensional? More specifically, if any Banach space is a Polish space?

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up vote 9 down vote accepted

Every separable Banach space is Polish by definition. For instance the sequence spaces $\ell^{p}$ for $1 \leq p \lt \infty$, the spaces of $p$-integrable functions $L^{p}[0,1]$ with $1 \leq p \lt \infty$ (see the Wikipedia article) or the space of continuous functions $C[0,1]$, or more generally $C(K)$ with $K$ compact metrizable (and infinite) with the supremum norm. All these spaces are clearly infinite-dimensional.

One can show that every separable Banach space embeds isometrically into $C[0,1]$, so this latter space is in some sense the universal separable Banach space.

On the other hand, it is not hard to show that the space $\ell^{\infty}$ of bounded sequences or the space $L^{\infty}[0,1]$ are not separable.

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By the way, Urysohn showed that there is a universal Polish space: There is a Polish space $U$ such that for every other Polish space $X$ there exists an isometric embedding $f: X \to U$. Googling for "Vershik, Urysohn space" should lead you to many articles on that fascinating space $U$ which is of course infinite-dimensional in any possible sense (since it contains the Banach spaces mentioned above). – t.b. May 31 '11 at 9:51
Well, as Polish spaces are special topological spaces rather than metric spaces, I should have said "complete separable metric space" instead of Polish throughout, but I'm used to confusing the two things whenever convenient. Sorry about this slight inaccuracy. – t.b. May 31 '11 at 11:09

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