Take the 2-minute tour ×  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?   R. Fry  share|improve this question  1 Answer 1
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.
share|improve this answer 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, – 