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How to construct the homeomorphism between $C[0,1]$ and $C[0,1]\setminus\{\theta\}$ in the explicit form?

Here, as usual, $C[0,1]$ is the Banach space of continuous functions $f:[0,1]\to\mathbb{R}$ with the norm $$ \|f\|=\max\limits_{x\in[0,1]}|f(x)|; $$ and $\theta(x)\equiv 0$ $\forall x\in[0,1]$.

Ilnara
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  • Interesting question, though maybe you should explain what motivated it. I wonder also whether $C[0,1]$ and $C[0,1] \setminus {\theta}$ could be isomorphic as posets. – Mike F Jun 15 '13 at 19:59
  • Maybe this is obvious, but what is the underlying topology? – parsiad Jun 15 '13 at 19:59
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    One way is to use the non-complete norm-technique as described by Bessaga-Pelczynski in Selected topics in infinite-dimensional topology, chapter III, §5: page 1 and page 2. You can make this as explicit as you wish. – Martin Jun 15 '13 at 20:13

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