TITLE: Are $(C[0,1],d_\infty)$ and $(C[0,1],d_1)$ homeomorphic?
QUESTION [5 upvotes]: Two metric spaces are said to be homeomorphic if there is a bijection f between them such that $f$ and  $f^{-1}$ are both continuous.
Consider $C[0,1]$ with metrics:
$d_\infty (f,g)=\max_{x\in [0,1]}|f(x)-g(x)|$
$d_1(f,g)=\int_0^1|f(x)-g(x)|dx$
We already know that the identity map $(C[0,1],d_1)→(C[0,1],d_∞)$ is not continuous (Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous). Does this imply $(C[0,1],d_∞)$ and $(C[0,1],d_1)$ are not homeomorphic?
Or could you find a bijection which is continuous in both direction?
Any help is appreciated.

REPLY [2 votes]: No, they are not. Confirming a conjecture of Banach, Victor Klee proved that if there is a complete metric on a normed (or, more generally, metrizable topological vector) space inducing the norm (vector space) topology then the norm (uniformity induced by the vector space topology) is complete. This can be seen, e.g., in Köthe's book topological Vector Spaces I, page 165.
Clearly, $d_1$ is not complete.