# Approximation Property

It seems to be a folk result that l_infinity has the approximation property, even the bounded approximation property, and also, I think, even the so-called propery pi (approximation property) of Lindenstrauss.

This is alluded to in a few texts, but I cannot seem to find the proof, which is presumably obvious.

Does anyone have a reference or an easy solution?

Cheers,

R. Fry

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## 1 Answer

Partition the measure space into finitely many sets and consider the span of their indicator functions. This space is isometrically isomorphic to $\ell_\infty^n$ for appropriate $n$ and hence is the range of a norm one projection. Index the partitions by refinement to get a net of norm one finite rank projections that converge strongly to the identity.

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Dear Prof Johnson, Thank you for the fast response. I am co-supervising a Masters student who needed an answer quickly so he could include it in his thesis which is to be handed in tomorrow! Cheers, –  Robb Fry Mar 11 '10 at 4:19