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a.$\{(z_1,z_2)\in \mathbb C\times \mathbb C:z_1^2+z_2^2=1\}$

b.the closure of the unit ball of $C^1[0,1]$ in $C[0,1]$

c.unit sphere in $\ell_2$

a. is not compact since it is not bounded.

c. is not compact since I can find $e_n\in \ell_2$ which will not have any convergent subsequence.Am I correct.But

b. I cant solve.How to do it?

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    $\begingroup$ $C^1[0,1]$ is infinite dimensional.The closed unit ball of infinte dimensional spaces is never compact!here same story for $\ell_2$. $\endgroup$ – BigM Dec 7 '14 at 3:58
  • $\begingroup$ can u please explain the reasons $\endgroup$ – Learnmore Dec 7 '14 at 4:10
  • $\begingroup$ It basically follows from Riesz's lemma. $\endgroup$ – BigM Dec 7 '14 at 4:14

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