Triangular inequality in norm?

Question

In $C[a,b]$, does the following norm next $\left \| f \right \|$ satisfy triangular inequality? $$\left \| f \right \|=\max\left \{ \sup\left | f \right |,\sup\left | f{}' \right | \right \}$$

I see that is true for cases $\max(f+g)\leq \max(f)+\max(g)$ and similary with $f'$ and $g´$. But $f$ and $g´$, and $f´$ and $g$?

Thanks.

Answer 1

Yes, the triangle inequality is still satisfied.

We note that the norm $\|\cdot\|^\ast$ given by $\|f\| = \sup|f|$ is well defined.

We also need to know the following relation: $$\max\{a+b,c+d\}\le\max\{a,c\}+\max\{b,d\}.$$

\begin{align}\|f+g\|&=\max\{\sup|f+g|,\sup|f'+g'|\} \\&= \max\{\|f+g\|^\ast,\|f'+g'\|^\ast\}\\&\le\max\{\|f\|^\ast+\|g\|^\ast,\|f'\|^\ast+\|g'\|^\ast\}\\&\le\|f\|+\|g\|\end{align}