Weak convergence in a reflexive Banach space implies poitwise convergence

Question

Let $X=W^{1,p}(\Omega)$ for $p>1$, which is a reflexive Banach space and $u_n\in X$ converges weakly to $u$ in $X$. Then does it follow that upto a subsequence $u_n\to u$ pointwise a.e. in $\Omega$.

I know that this is true, but want to know does it follow only from the REFLEXIVITY of $W^{1,p}(\Omega)$?

If so, how?

Can you kindly explain?

Thanking you.

Answer 1

Take $S:=\{f_n(t)=e^{int}\}\subseteq L^2(\mathbb T).$ Then, if $p$ is a polynomial, $\langle f_n,p\rangle\to 0$ and since the polynomials are dense, it follows that $f_n\rightharpoonup 0$. But $S$ has no convergent subsequence.