Weak-* convergence in $L^{\infty}$ implies weak convergence in $L^p$ on bounded set

Question

In a lecture I found the following result: "We remark that when $\Omega$ is bounded the weak-* convergence of $u_{n}$ in $L^{\infty}(\Omega)$ to some $u \in L^{\infty}(\Omega)$ implies weak convergence of $u_{n}$ to $u$ in any $L^p$, $1 \leq p < \infty$." Does anyone know a book or some other source where this result is proved?

Answer 1

I don't know of a reference but that might be because the result isn't so hard to see. Since $\Omega$ is bounded, we have that $L^q(\Omega) \subseteq L^{1}(\Omega)$ for every $1 \leq q \leq \infty$.

Now weak-$*$ convergence in $L^\infty(\Omega)$ means that for every $v \in L^1(\Omega)$, $$\int u_n v dx \to \int u v dx$$ as $n \to \infty$. To prove the weak convergence in $L^p(\Omega)$ we need to prove that if $v \in L^q(\Omega)$ where $p^{-1} + q^{-1} = 1$, then $\int u_n v \to \int u v$. But since $L^q(\Omega) \subseteq L^1(\Omega)$, this is immediate by the weak-$*$ convergence in $L^\infty$.