Let $(V,\|\cdot\|)$ and $(W,\|\cdot\|')$ be a normed vector spaces. We say $V$ and $W$ are isomorphic if there exists a map $L : V \to W$ such that $L$ is both a linear bijection and a homeomorphism on the norm topologies. Say $V$ and $W$ are algebraically isomorphic if there exists a linear bijection $T : V \to W$. ($T$ not necessarily a homeomorphism.)
Suppose $V$ and $W$ are both algebraically isomorphic and homeomorphic in the norm topology. Are $V$ and $W$ isomorphic? In other words, if there exists a linear bijection $T : V \to W$ and a homeomorphism $H : V \to W$, does there necessarily exist a linear bijective homeomorphism $T : V \to W$?
Any two separable infinite dimensional Banach spaces are homeomorphic by a (very deep) theorem of Kadetz and every separable infinite dimensional Banach space has a basis with the cardinality of the continuum by a result of Mackey. So every two separable infinite dimensional Banach spaces that are not isomorphic are counterexamples.