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 $p<\infty$.