Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space . I know that $C^1([a,b])$ is a Banach space with the $C^1$ norm. But is it a Banach space with the usual norm $||f||_\infty=sup|f|$ ? I think it shouldn´t be, but when trying to prove it I always get that it is. Any idea of a Cauchy sequence in $C^1[0,1]$ which is not convergent ?
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1See this. – David Mitra Jan 07 '21 at 09:55
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2the "usual norm" for $C^1$ is of course, the $C^1$ norm. – Calvin Khor Jan 07 '21 at 09:56
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2Approximate $\sqrt x$ on $[0,1]$ by a sequence of polynomials. – Kavi Rama Murthy Jan 07 '21 at 09:56
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Thank you David, I didn´t found that question. – Indiano28 Jan 07 '21 at 09:57