What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$?

Question

$L^2(\mathbb R)$ is isomorphic to $\ell^2(\mathbb R)$ (which has the cardinality of $\mathbb R$ since there is an injection to the space of continuous functions which has the cardinality of $\mathbb R$), but what about different $1 \le p < \infty$?

What happens if we consider $L^p(O)$, where $O \subseteq \mathbb R$ is open?

Answer 1

@PhoemueX gave the answer above (see https://en.wikipedia.org/wiki/Separable_space#Cardinality; separable normed spaces have at most cardinality of $\mathbb R$). For a proof of that statement see the comment of Nate Eldredge's to the post.