Let $S^n = \{ x \in \mathbb R^n : ||x|| = 1 \}$ be the unit sphere, then there exists no bijection between $S^n$ and an open subset of any Banach space.
How to show that?
I see that $S^n$ could not be homeomorphic to any Banach space, as $S^n$ is compact, and any Banach space is not compact as for example the sequence $\{ n\cdot u \}$ for any $u \ne 0, u \in E$ does not has a limit. But for the more general notion of bijectivity I do not see it...
This is not true, the cardinal of $S^1$ is $R$ since $S^1$- one point is $R$, so there exists a bijection between $S^1$ and $R$ and $R$ is a Banach space.