Take the 2-minute tour ×  Let $E$ be a closed subspace of
$L^2[0,1]$. Suppose that $E\subset{}L^\infty[0,1]$. Is it true
that $E$ is finite dimensional?  PS. This is actually a question
from the real analysis qualifier. I came across it as I was
teaching qualifier preparation course, and was solving problems
from old qualifiers. So, though it might follow from some
advanced theory of Banach spaces, I am most interested in the
'elementary' solution, using only methods from standard real
analysis course. Note: if $E\subset{}C[0,1]$, then it is a
problem from Folland, and there is a solution there. However, it
does not work for $L^\infty$, not without some trick.
share|improve this question Sorry, I left a reference (to
Rudin's book) as an answer without properly reading your
question, and in particular the fact that you wanted a proof
avoiding functional-analytic tricks –  Yemon Choi Jan 19 '11 at
19:12 Since it seems that using the open mapping theorem is
allowed, perhaps the reference to Rudin might still be