Intersection of two $L^p$ spaces is complete.

Question

Given $1 \leq p < q < \infty$, I wish to show that $L^p \cap L^q$ is a Banach space given the norm $\|f\| = \|f\|_p + \|f\|_q$. I have shown that this space is a vector space and that the norm is indeed a norm, all I have left is showing that this space is complete and am have some trouble. I was able to deduce that if a sequence is Cauchy in $L^p \cap L^q$ then it must be Cauchy in $L^p$ and $L^q$, but I am not sure where to go from here. Any advice would be appreciated.

Answer 1

If $(f_n)$ is Cauchy it converges in $L^{p}$ to some $g$ and in $L^{q}$ to some $h$. There is a subsequence $(f_{n_i})$ which converges to $g$ almost everywhere and there is a further subsequence which converges to $h$ almost everywhere. It follows that $g=h$ almost everywhere. It is now obvious that $f_n \to g$ in the given norm.

Answer 2

It suffices to show that given a sequence $(f_n)_n$ in $L^p \cap L^q$ such that $\sum_n \lVert f_n \rVert < \infty$, the sequence $(\sum_{k=1}^n f_k)_n$ is convergent in $L^p\cap L^q$. (see Theorem 5.1 of Folland's Real Analysis)

Let $(f_n)_n$ be a sequence in $L^p\cap L^q$ such that $\sum_n \lVert f_n \rVert < \infty$. Due to the Monotone Convergence Theorem, there exists a function $g$ defined a.e. with values in $[0,\infty]$ such that $g=\sum_n \lvert f_n \rvert$. $\lVert \cdot \rVert$ is a sum of norms, hence a norm. A limit process with the triangle inequality assures that $g \in L^p \cap L^q$.

In this situation, we claim that there exists an a.e. defined function $f$ such that $f=\sum_n f_n$. Indeed: for a.e. $x$, $\left(\sum_{k=1}^n \lvert f_k(x) \rvert \right)_n$ is a Cauchy sequence, then so is $\left(\sum_{k=1}^n f_k(x) \right)_n$. $g \in L^p\cap L^q$, so $f \in L^p \cap L^q$.