# What could the notation $l^\infty(\mathcal{F})$ mean, where $\mathcal{F}$ is a set of measurable functions?

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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Will you please give a precise reference to where this is used? – Jonas Meyer Nov 12 '10 at 23:32
"Weak convergence and Empirical Processes", Aad W. van der Vaart Jon A. Wellner, page 81 (2.1). Thanks. – normvector Nov 12 '10 at 23:36
Thanks for the reference. Here's a link to Google Books for those interested: books.google.com/… – Jonas Meyer Nov 12 '10 at 23:47
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 $\mathcal{F}\to \mathbb{R}$. – Willie Wong Nov 13 '10 at 0:16