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?
Polish spaces: finite or infinite?
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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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1Well, 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. – 2011-05-31