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In the book Weak convergence and Empirical Processes, by Aad W. van der Vaart and Jon A. Wellner, on page 81, the notation $l^\infty(\mathcal{F})$ appears, where $\mathcal{F}$ is a set of measurable functions, $f \in \mathcal{F}$ then $f \colon \mathcal{X} \rightarrow \mathbb{R}$.

I am not sure what $l^\infty(\mathcal{F})$ is. I know what $l^\infty$ is by itself, the set of bounded sequences (Wikipedia), but not sure about this one.

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    I'm not 100% sure, but the fact that $\mathcal{F}$ is mentioned as an "indexing set" suggests to me that it actually means just that: the everywhere bounded functions from $\mathcal{F}\to \mathbb{R}$.2010-11-13

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Did you check page 34 (which I got by checking the List of Symbols on page 506)?

http://books.google.com/books?id=seH8dMrEgggC&pg=PA34

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