Consider $X=C^{1}([-1,1])$, the space of real valued $C^1$ functions defined on the interval $[-1,1]$. Define a norm on $X$ by $||f||:=\sup_{x\in[-1,1]}(|f(x)|+|f'(x)|)$. Consider the set $Y:=\{g:||g||\leq 1\}\subset X$.
My question is, is $B$ compact? By Arzela-Ascoli, in order to show that $B$ is compact, it suffices to show that it is bounded (which it is) and equicontinuous. I am kind of lost on the equicontinuity part (I am inclined to believe that it is not equicontinuous, but I cannot think of a function that justifies my claim...), so any help will be useful.
Thanks.
Arzela-Ascoli Theorem does not give compactness w.r.t. the given norm. It gives compactness only w.r.t the sup norm.
The closed unit ball of an infinite dimensional normed linear space is never compact! So $Y$ is not compact.