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| -rw-r--r-- | content/analysis_3.tex | 17 |
1 files changed, 16 insertions, 1 deletions
diff --git a/content/analysis_3.tex b/content/analysis_3.tex index bd799a7..ae7cf5f 100644 --- a/content/analysis_3.tex +++ b/content/analysis_3.tex @@ -528,4 +528,19 @@ Sei $D \subseteq \R^m$ offen und beschränkt mit dünnsingulärem $C^1$-Rand, $f $$\int_D div f(x) dx = \int_{\partial D} (f(x)|\nu(x)) d\sigma(x)$$ -Mit $div f(x) := spur f'(x) = \partial_1 f_1(x) + \cdots + \partial_m f_m(x)$. +Mit $\text{div} f(x) := \text{spur} f'(x) = \partial_1 f_1(x) + \cdots + \partial_m f_m(x)$. + +\section*{Lebesguesche Räume} + +Für messbare $f : X \to \overline\R$: + +\vspace{-4mm} +\begin{align*} +\|f\|_p &= \left(\int_X |f|^p d\mu\right)^\frac{1}{p} \text{ für } p \in [1,\infty)\\ +\|f\|_\infty &= \text{esssup}_{x \in X} \|f(x)\|\\ + &= \inf\left\{ c > 0 | \exists \text{ NM } N_c : \forall x \in X \setminus N_c : |f(x)| \leq c\right\} +\end{align*} + +\subsection*{$L^p$-Räume} + +$$\L^p(X,\A,\mu) := \{ f : X \to \R | f \text{ mb.}, \|f\|_p < \infty\}$$ |