Cardinality of space of all funcions $f: \mathbb R \rightarrow \mathbb R$ is $\beth_2$. However, cardinality of space of all such square-integrable functions, space $L^2$, is $\beth_1=\mathfrak c$, equal to cardinality of $\mathbb R^n$.
So, from set-theoretical point of view, $L^2$ is as big as usual manifolds, which motivates the question:
What are the topological properties of $L^2$ space (with usual topology) and, more precisely, is there any manifold which is homeomorphic to $L^2$?