Weirdly defined ball compact in $C^1([0, 1])$

Question

Consider$$B := \left\{u \in C^2([0, 1]) : \sum_{i=0}^2 \sup_{x \in [0, 1]} \left|u^{(i)}(x)\right| \le 1\right\}$$as a subset of $C^1([0, 1])$. How do I see that it is compact in $C^1([0, 1])$?

Answer 1

It isn't. You can construct a sequence of such functions that converges to another function (outside $B$) in the norm of $C^1[0,1].$

Start with the block function $f$ that is $0$ on $[0,1/2)$ and $1$ on $[1/2,1]$. Don't worry that this function is outside all of the spaces under consideration; suffice that it is bounded.

Consider a sequence $(f_n)_n$ of nonnegative continuous (not necessarily differentiable) functions that converges pointwise to $f$ from below, i.e., $0\leq f_n(x)\leq f_{n+1}(x)\leq f(x).$

Now the double primitive functions of the $f_n$ all belong to $B$ and they converge to the double primitive function of $f$ in the norm of $C^1[0,1].$

So we have an infinite sequence in $B$ that does not have any convergent subsequences (in $B$) in the norm of $C^1[0,1];$ therefore $B$ cannot be compact in the norm of $C^1[0,1].$