It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In fact, the map $f(x)\mapsto sgn(f(x))|f(x)|^{q/p}$ provides an explicit homeomorphism $L^p(\Bbb R) \to L^q(\Bbb R)$.


However this argument cannot be applied to the case where $p$ or $q$ equals infinity.
Thus, I'm asking whether there is a homeomorphism from $L^\infty(\Bbb R)$ to any (and hence every) $L^p(\Bbb R)$ with $p<\infty$.



Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?
Paul is right. $L^2(\mathbb{R})$ is separable. (The rational simple functions ought to be one example of something ctbl. and dense.) 


However, $L^{\infty}(\mathbb{R})$ isn't separable. By Jones' Lemma, if $L^{\infty}(\mathbb{R})$ were separable then any closed discrete (i.e., nonclustering) set must be of size less than continuum. But the characteristic functions $\chi\_{[0,x]}$, $x\in[0,\infty)$, are pairwise of distance 1 from eachother.


