Hello everyone, I'm completly new to this topic, so I don't have any clue on this
Let \(\displaystyle A \subset \mathbb{R}^n\) and \(\displaystyle f \in C^l(\overline{A})\)
Show that \(\displaystyle \ ( C^l(\overline{A}) , \ \| f\|_{l,\infty,A} )\) is a banach space, where
\(\displaystyle \| f\|_{l,\infty,A} := \max_{|d| \le l } \sup_{x \in A} | \partial^d f(x) |\)
All I know is that I need to show: \(\displaystyle ( C^l(\overline{A}) , \ \| f\|_{l,\infty,A} )\) is complete, e.g. every cauchy sequence convergences
If (f_n) is a cauchy sequence in C^l, why does a f in C^l exist with f_n converges to f?
I don't get it, so can someone help me out of this problem, pls? This would be great
Thank you
Let \(\displaystyle A \subset \mathbb{R}^n\) and \(\displaystyle f \in C^l(\overline{A})\)
Show that \(\displaystyle \ ( C^l(\overline{A}) , \ \| f\|_{l,\infty,A} )\) is a banach space, where
\(\displaystyle \| f\|_{l,\infty,A} := \max_{|d| \le l } \sup_{x \in A} | \partial^d f(x) |\)
All I know is that I need to show: \(\displaystyle ( C^l(\overline{A}) , \ \| f\|_{l,\infty,A} )\) is complete, e.g. every cauchy sequence convergences
If (f_n) is a cauchy sequence in C^l, why does a f in C^l exist with f_n converges to f?
I don't get it, so can someone help me out of this problem, pls? This would be great
Thank you