Is $l^p$ closed in $L^p$?

Question

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a few things that would suggest that this is true, but I don't see how this could be shown?

Answer 1

it is a topological one.

In that case, $X$ must be a closed subspace. That is, because an isomorphism of topological vector spaces is not only a topological isomorphism, but also an isomorphism in the category of uniform spaces (with uniformly continuous maps as morphisms), so if $X$ (with the subspace topology induced by $L^p$) is isomorphic to $l^p$ as a topological vector space, it is complete, because $l^p$ is. And a complete subspace of a Hausdorff uniform space is closed.