If $u_n\to u$ weakly in $L^p$ show that $\|u_n\|\leq C$ for a certain $C>0$.

Question

If $u_n\to u$ weakly in $L^p$ show that $\|u_n\|\leq C$ for a certain $C>0$.

Let denote $q$ the conjugate of $p$, i.e. $\frac{1}{p}+\frac{1}{q}=1$.

I tried to majorate $\|u_n\|_{L^p}$ by something like $$K+\left|\int (u_n-u)\varphi\right|$$ for $\varphi\in L^q$ but with no success...

Answer 1

It is a consequence of the more general fact:

If $(X,\Vert \cdot \Vert)$ is a Banach space and $\{x_n\} \subset X$ is a sequence converging weakly to $x$ in $X$, then $\{ \lVert x_n \rVert \}$ is bounded.

Indeed, from the definition of weak convergence we have that $\{\langle f,x_n\rangle\}$ is a bounded subset of $\mathbb{R}$ for every $f \in X'$. Then it suffices to apply the uniform boundedness principle (aka, Banach-Steinhaus theorem) to obtain the result.

References: Proposition 3.5 of Brezis' book Functional Analysis, Sobolev Spaces and Partial Differential Equations or Theorem 1 in chapter V of Yosida's Functional Analysis.