The closed unit ball is not compact in infinite dimension spaces. Why? We know that in finite dimension spaces the closed unit ball is compact, that is if H is a finite dimension space, then there exists an $u$ in the closed unit ball in H and $T \in \mathcal{L}(H, \mathbb{R})$ such that $||T|| = |T u|$. 
Why doesn't it happen in infinite dimension spaces? 
For example, let $H = C([a, b])$ with the sup norm. Why doesn't exist an element $u$ in the closed unit ball in $C([a, b])$ such that $T u = ||T||$?
Thank you!

Essentially because sequences can wander off to infinite. Eg take any function with sup norm $1$ supported in the interval $[0,1].$ 
Call it $f.$ 
Define $f_n(x) = f(x-n).$ 
These functions have disjoint support and their distance in the sup norm is $1.$
