If a subspace of $L^p\cap L^q$ is closed with respect to both norms, the norms are equivalent on this subspace

Question

Let $\mu$ be a positive measure, $1< p,q<\infty$ and let $X$ be a linear subspace of $L^p(\mu)\cap L^q(\mu).$ Suppose $X$ is closed in $L^p(\mu)$ and also $X$ is closed in $L^q(\mu)$. Prove that $\|.\|_p$ and $\|.\|_q$ are equivalent on $X$.

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Truly speaking I have no idea how to start. I know I have to find the some constant that that will show the equivalent on the norms.

Answer 1

Expanding the comment by David Mitra: the identity map $I:(X,\|\cdot\|_p)\to (X,\|\cdot\|_q)$ has the graph $$\{(x,x) : x\in X\}\subset L_p\times L_q$$ which is a closed set by the assumption. Since both $(X,\|\cdot\|_p)$ and $ (X,\|\cdot\|_q) $ are Banach spaces, the Closed Graph Theorem shows that $I$ is a bounded operator.

The same applies with $p$ and $q$ interchanged; thus $I$ is an isomorphism.