# Find an isomorphism between $C[0,1]$ and a proper subspace of itself

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 $$E$$.

• What's $C[0,1]$ here -- continuous functions supported over $[0,1]$? – Jack Crawford Jun 9 at 13:59
• yes C[0,1] is the continuous functions over [0,1] – user680813 Jun 9 at 14:17

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.

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