Is $C^1[0,1]$ linearly homeomorphic to $C[0,1]$ ?
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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, onevisit wonder: Is this your homework? 

