Functional analysis proof for $L^p$ spaces and Schur property.

Question

It is easy to construct an example in $L^2(\Omega)$ (where $\Omega$ is a bounded box in $\mathbb{R}^n$) for which weak convergence holds but strong does not. My question is: is there a general property of the $L^p$ spaces or a purely functional analytic proof of this fact?

Answer 1

Note that $L_p(\Omega)$ contains a subspace isomorphic to $\ell_2$. For example, take the closed span of a sequence of independent, symmetric, two-valued random variables (and use Kchinchine's inequality). Then you may use your favourite proof for $\ell_2$.