# Pick out compact sets

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?

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