I have to prove that the following norms are equivalent in $C^1([0,1])$: $$ || f ||_* = |f(0)| + \max_{t \in[0,1]} |f'|\\ || f||_\infty = \max_{t \in[0,1]} |f| $$
I have: \begin{align} || f||_\infty &= \max_{t \in[0,1]} |f| \\&= \max_{t \in[0,1]} \left|f(0) + \int_{0}^{t}f'(\tau)d\tau\right| \\ &\le |f(0)| + \int_{0}^{t}|f'(\tau)|d\tau \\&\le
|f(0)| + \int_{0}^{1} |f'(\tau)|d\tau \\&\le |f(0)| + \max_{t \in [0,1]}|f'(t)| \\&= ||f||_*\end{align}
Now I'm in trouble to find the other inequality.
Another little question. I know that $ ||f||_\infty $ and $ ||f||_1 $ are not equivalent in $C([0,1])$ but are they equivalent in $C^1([0,1])?$ I'm asking since on some notes I'm asked to prove that the $||f||_*$ is equivalent to both these two norms, but it may be an error.
Best Answer
You cannot prove it, since it is false. If $f_n(x)=\sin(nx)$, then $\lVert f_n\rVert_\infty\leqslant1$, whereas $\lVert f_n\rVert_*=n$.
The same sequence of functions can be used to prove that the norms $\lVert\cdot\rVert_\infty$ and $\lVert\cdot\rVert_1$ are not equivalent in $\mathcal C^1\bigl([0,1]\bigr)$.