I have found this exercise about the closed graph theorem, but I am not yet familiar with the vector spaces of locally integrable functions. So I need your help.
The aim of the exercise is to show that any inclusion between two complete vector subspaces continuously embedded in $L_{loc}^{1}\left(\Omega\right) $ is in fact a continuous embedding (in the sense that the canonical injection is continuous).
More precisely, consider an open set $\Omega$ and consider two vector subspaces $E$ and $F$ of the space $L_{loc}^{1}\left(\Omega\right) $. We suppose that $E$ and $F$ are endowed with norms $\left\Vert {.}\right\Vert _{E}$,$\left\Vert {.}\right\Vert _{F}$, which make them complete and such that any sequence $\left( f_{n}\right) _{n}$ which converges to $0$ in $E$ (resp. $F$) converges to $0$ in $L_{loc}^{1}\left(\Omega\right) $ in the following sense : for any $\varphi \in C_{0}^{\infty }\left(\Omega\right) $, $\int f_{n}\left( x\right) \varphi \left( x\right) dx\underset{% n\rightarrow +\infty }{\longrightarrow }0$
Suppose that $E\subset F$. Prove that there exists $c>0$ such that $% \left\Vert u\right\Vert _{F}\leq c\left\Vert u\right\Vert _{E}$ for all $% u\in E$. (Hint: use the closed graph theorem).
The graph of the inclusion map is $\{(f,f)|f\in E\}$. We want to prove that the graph is closed. So, assume $(f_n, f_n)$ is a sequence in the graph that converges to some point $(f,g)\in E\times F$ in the product norm, we need only prove that $f=g$.
Use the given condition, both $(f_n-f)$ and $(f_n-g)$ converge to $0$ in the topology for $L^1_{loc}(\Omega)$, which implies that $f=g$.