Check that formulas define norms

Question

So I have task in which I have to prove that the following formulas define norms in $C^k[0,1]$:

(a) $\|f\|_1=\max_{s=0,1,...,k} \sup_{0\le t\le 1}|f^{(s)}(t)|$

(b) $\|f\|_2=\sup_{0\le t\le 1}|f^{(k)}(t)| + \max_{j=0,1,...,2k}|f(\frac{j}{2k})|$

In both cases I have problem with triangle inequality. I'm not sure anything..

eg. $\|f+g\|_1= \max_{s=0,1,...,k} \sup_{0\le t\le 1}|(f+g)^{(s)}(t)|=\max_{s=0,1,...,k} \sup_{0\le t\le 1}|(f^{(s)}+g^{(s)})(t)| \le \max_{s=0,1,...,k} \sup_{0\le t\le 1}(|f^{(s)}(t)| + |g^{(s)}(t)| ) \le \|f\|_1 + \|g\|_1 $

Please help me to better understand it.

Answer 1

Hint: Given $A$ and $B$ two functions in $C^k([0,1],\mathbb{R})$, then we have the following:

\begin{align*} \sup_x (|A(x)| + |B(x)|) \leq \sup_x |A(x)| +\sup_x |B(x)|\\ \max_x (|A(x)| + |B(x)|) \leq \max_x |A(x)| +\max_x |B(x)| \end{align*}