Prove the space $L^p(X) \cap L^q(X)$ with the norm $||f||_{L^p \cap L^q}=||f||_p+||f||_q$ is a Banach space

Question

$X$ is a space with positive measure and $1\le p<q\le +\infty$. I have to prove that $L^p(X) \cap L^q(X)$ is a complete space i.e. every Cauchy's sequence converges in this space with the norm $\lVert f\rVert_{L^p \cap L^q}=\lVert f\rVert_p+\lVert f\rVert_q$. Can I use the fact that if I have a sequence in $L^p$ that converges a $f \in L^P$ then exist a subsequence $f_{n_{k}}$ that converges a.e?

Answer 1

You know that both $L^p$ and $L^q$ are Banach spaces. Fix a Cauchy sequence $(f_n)_{n\geq 1}$ in $L^p\cap L^q$.

• Since $(f_n)_{n\geq 1}$ is a Cauchy sequence in $L^p$, it converges in $L^p$. Let $f\in L^p$ be the limit.

• Since $(f_n)_{n\geq 1}$ is a Cauchy sequence in $L^q$, it converges in $L^q$. Let $g\in L^q$ be the limit.

You now have to show that $f=g$. If that's true, then you're done (can you see why?). To do that, use whatever tool you want/can: for instance, you can use the fact that $L^p$ convergence implies convergence in measure, and use uniqueness of that limit.