Is the vector space complete under this norm?

Question

We have $C^2[0,1]$ under the norm: $\|f\|=\|f\|_\infty + \|f''\|_\infty$. I know that it shouldn't be complete because we have no information about first derivative but I'm struggling to find an example which shows that. Also a general proof that $C^k[0,1]$ is complete iff all the derivatives are included in the norm would be great.

Answer 1

You can find constants $A$ and $B$ such that $\|f'\|_{\infty}\le A\|f\|_{\infty}+B\|f''\|_{\infty}$, by using the following \begin{align} (b-a)f'(x)&=(b-x)f'(x)+(x-a)f'(x) \\ &= -\int_{x}^{b}\frac{d}{dt}\{(b-t)f'(t)\}dt+\int_{a}^{x}\frac{d}{dt}\{(t-a)f'(t)\}dt \\ &=\int_{x}^{b}\{f'(t)-(b-t)f''(t)\}dt+\int_{a}^{x}\{f'(t)+(t-a)f''(t)\}dt \\ &=f(b)-f(x)-\int_{x}^{b}(b-t)f''(t)dt+f(x)-f(a)+\int_{a}^{x}(t-a)f''(t)dt \\ &=f(b)-f(a)+\int_{a}^{x}(t-a)f''(t)dt-\int_{x}^{b}(b-t)f''(t)dt. \end{align}