Is ${C}[0,1],\Bbb{R}$ homeomorphic to any $\Bbb{R^n}$, for an integer $n$?

Question

The norm in ${C}[0,1],\Bbb{R}$ is the norm of $L^1$

Answer 1

Hint: Is the space still path-connected after removing one element?

Answer 2

No. There are embeddings of $\mathbb R^k$ into $C([0,1], \mathbb R)$ with $L^1$ norm for all positive integers $k$: all you need is to take $k$ linearly independent continuous functions $f_j$ on $[0,1]$, and map $(t_1, \ldots, t_k) \mapsto t_1 f_1 + \ldots t_k f_k$. But there is no embedding of $\mathbb R^k$ into $\mathbb R^{n}$ for $k > n$.