How are $C^0,C^1$ norms defined

Question

How are $C^0,C^1$ norms defined? I know $L_p,L_\infty$ norms but are the former defined.

Answer 1

On $\mathcal{C}^{0}([a,b])$, the usual norm is

$$ \Vert f \Vert = \sup \limits_{x \in [a,b]} \vert f(x) \vert $$

(the interesting point is that $\left( \mathcal{C}^{0}([a,b]), \Vert \cdot \Vert \right)$ is a Banach space.

On $\mathcal{C}^{1}([a,b])$, you can define the norm

$$ \Vert f \Vert_{\mathcal{C}^{1}} = \sup_{x \in [a,b]} \vert f(x) \vert + \sup_{x \in [a,b]} \vert f'(x) \vert $$

(and $\left( \mathcal{C}^{1}([a,b]),\Vert \cdot \Vert_{\mathcal{C}^{1}} \right)$ is also a Banach space.)

Answer 2

Let $\Omega$ be a domain.

Set $$[u]_{k,0,\Omega} = \sup_{|\beta| =k}\sup_{\Omega}|D^{\beta}u|$$

Then, $$||u||_{C^{k}(\bar{\Omega})} = \sum_{j=0}^k{[u]_{j,0,\Omega}}$$