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I had shown that unit ball of $C^1[0,1]$ is compact with norm $||f||_1=||f||_{\infty}+||f'||_{\infty}$ using Arzela Ascoli thoerem.

Compactness implies Totally bounded.

Is there is any other way to directly prove this fact that unit closed ball is totally bounded?

Please give me a hint

Any Help will be appreciated

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Your space is an infinite dimensional normed linear space. The unit ball of such a space is never compact. What Arzela - Ascoli theorem tells you is that if $(f_n)$ is in the unit ball of this space then there is a subsequence which converges in the sup norm. But this does not give convergence in the norm of $C^{1}[0,1]$.

The unit ball of this space is also not totally bounded.

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