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Is $C^1[0,1]$ linearly homeomorphic to $C[0,1]$ ?

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closed as too localized by Andrés Caicedo, Yemon Choi, Harry Gindi, Wadim Zudilin, Bill Johnson Jan 11 '11 at 18:47

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Mr Smith, you definitely have Slavic roots: "gomeomorphic" gives yourself completely... Can't you put more effort on your question? It sounds a homework. – Wadim Zudilin Jan 8 '11 at 12:48
Dear Mr Smith, Wadim was merely suggesting what you could have already read here, – Gjergji Zaimi Jan 8 '11 at 13:07
Mr Smith, I have merged your two user accounts. If you wish to respond to someone else's answer, please leave a comment instead of creating a new answer. Please try to be civil. – S. Carnahan Jan 8 '11 at 13:30

By integration, $C[0,1]$ is linearly homeomorphic to a hyperplane in $C^1[0,1]$. For example, all elements $g$ of $C^1[0,1]$ with $g(0)=0$. All classical Banach spaces are linearly isomorphic to their hyperplanes. In this case it can be proved constructively.

So, Mr Smith, one-visit wonder: Is this your homework?

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"It is believed that all Banach spaces are linearly isomorphic to their hyperplanes."? Banach's Hyperplane Problem was solved into negative by W.T.Gowers in 1993. – Ady Jan 8 '11 at 14:38
thanks, edited. – Gerald Edgar Jan 8 '11 at 16:53

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