I want to prove that if $E$ is a closed linear subspace of $C[0,1]$ and the elements of $E$ are functions of class $C^1$, then $E$ is finite dimensional. I have tried to prove that the unit ball in $E$ is compact, but I don't know how to do it. Can somebody help me? Thank you very much!
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What is $E$? Just a subset? – Sujit Bhattacharyya Jun 30 '19 at 13:50
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Sorry, it's a linear subspace. I forgot it. – JN_2605 Jun 30 '19 at 14:32
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@JN_2605 What is your norm? – Raito Jun 30 '19 at 14:56
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The norm of the maximum $||f||=\max\limits_{x\in[0,1]}|f(x)|$ – JN_2605 Jun 30 '19 at 15:36