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Let $E=C[0,1]$, the vector space of all continuous functions defined on $[0,1]$, equipped with the sup norm. Find an isomorphism between $E$ and a proper subspace of $E$.

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  • $\begingroup$ What's $C[0,1]$ here -- continuous functions supported over $[0,1]$? $\endgroup$ – Jack Crawford Jun 9 at 13:59
  • $\begingroup$ yes C[0,1] is the continuous functions over [0,1] $\endgroup$ – user680813 Jun 9 at 14:17
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Let $A$ be the subspace of $C[0,1]$ for which funcions are constant on $[\frac{1}{2},1]$. I think that the map $\phi\colon C([0,1])\to A$ defined by $\phi(f)(x)=f(\frac{x}{2})$ if $x\in[0,\frac{1}{2})$ and $\phi(f)(x)=f(1)$ if $x\in [\frac{1}{2},1]$ should work.

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  • $\begingroup$ Your function is not an isomorphism as it is not injective. For instance let $c_0$ denote the constant 0 function. Then $\phi(0) = \phi(\sin)$. $\endgroup$ – skullph Jun 9 at 14:12
  • $\begingroup$ Ups sorry, I forgot it shoud be an isomorphism. I'll edit it and write an isomorphism. $\endgroup$ – elescararriba Jun 9 at 14:17

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