5
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ In $C^0[a,b]$ one usually uses the $\sup$ norm, $$\lVert f-g\rVert_\infty=\sup_{x\in [a,b]}|f(x)-g(x)|$$ Are you meaning to ask about $C^0[a,b]$? $\endgroup$
    – Pedro
    Aug 8, 2013 at 16:05
  • $\begingroup$ @PeterTamaroff, I think so. How about $C^1$ $\endgroup$
    – Vaolter
    Aug 8, 2013 at 16:08

2 Answers 2

12
$\begingroup$

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.)

$\endgroup$
2
$\begingroup$

Let $\Omega$ be a domain.

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

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.