Prove that the $L^p$ and $L^q$ norms are equivalent on a closed subspace $X$ of $L^p(\mu) \cap L^q(\mu)$

Question

I was looking for a few clarifications on a post here on stack exchange, namely this one

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

My questions stem from this answer given: https://math.stackexchange.com/a/1478213/966646

Here are my questions:

1. Does $\mu$ need to be of positive measure, or could $\mu$ just be an arbitrary measure?

2. How does knowing that $I$ is bounded help prove the isomorphism?

3. Once we get that $I$ is an Isomorphism, how can we conclude that the norms in question are equivalent?

Answer 1

1. If $\mu$ is not a positive measure, then $\int \vert u\vert^p\mathrm{d}\mu$ can be negative and $\Vert u\Vert_p=\left(\int \vert u\vert^p\mathrm{d}\mu\right)^{1/p}$ is not well-defined.

2. $I$ is the identity map, as such it is automatically bijective and linear. It only remains to show that it is continuous or, equivalently, bounded.

3. If $I$ is bounded, this means that there is a constant $C>0$ such that $\Vert u\Vert_q=\Vert Iu\Vert_q\leq C\Vert u\Vert_p$. Since $p$ and $q$ are interchangeable, this implies the equivalency of the norms.