I want to prove weak compactness of $L^p$ when $p \neq 1,\infty$ and if $\sup_n \|f_n\| < \infty$, which is a problem in Stein and Shakarchi's Functional Analysis book. I've done the part where strong convergence implies weak convergence, and when $\sup_n \|f_n\| < \infty$, it is enough to check weak convergence on dense subset of functions $g$ in $L^q$ where $q$ is the conjugate of $p$.
Now I guess I need to show that if $\sup_n \|f_n\| < \infty$, there is $f \in L^p$ and subsequence $\{f_{n_k}\}$ such that $f_{n_k}$ converges weakly to $f$ i.e. $$ \int f_{n_k} g \,d\mu \to \int fg \,d\mu$$ for every $g$ in $L^q$. But I have been stuck on this for a few hours and have no idea where to start.
Can anyone help me out? Thanks!
This follows from the Banach-Alaoglu theorem. Basically you're trying to prove that the unit ball in $L^p$ is weakly compact (just rescale the norms of the $f_n$'s to see that). The crucial fact is that $L^p$, for $p \in (1,\infty)$, is reflexive, and reflexive Banach spaces have weakly compact balls.
Here is an outline of the proof:
If $X$ is reflexive, then the natural embedding $X \to X^{**}$ is an isomorphism, and so it is between the unit balls $B$ and $B^{**}$. Note that this is a homeomorphism (with weak topology on $B$ and weak* toplogy on $B^{**}$). Applying Banach-Alaoglu's theorem to $X^*$, we get that $B^{**}$ is weak* compact, and so is any space homeomorphic to it, in particular $B$ with the weak topology.
Note that this is not a full proof (I haven't justified many steps).