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authorAdrian Kummerlaender2017-03-20 21:22:32 +0100
committerAdrian Kummerlaender2017-03-20 21:22:32 +0100
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Add section on Riesz-Fischer's theorem
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@@ -658,3 +658,11 @@ Seien $f, g \in \L^p(\mu)$. Dann gilt $f + g \in \L^p(\mu)$ und:
\vspace{-2mm}
$$\| f + g \|_p \leq \|f\|_p + \|g\|_p$$
+
+\subsection*{Satz von Riesz-Fischer}
+
+Sei $1 \leq p < \infty$, $(f_n)$ Cauchyfolge in $\L^p(\mu)$ bzgl. $\|\cdot\|_p$.
+
+Dann existieren $f, h \in \L^p(\mu)$ und Teilfolge $(f_{n_j})_j$ s.d. diese f.ü. gegen $f$ strebt, $\forall j \in \N : |f_{n_j}| \leq h$ f.ü. gilt und $\displaystyle\lim_{n \to \infty} \|f_n - f\|_p = 0$ gilt.
+
+$L^p(\mu)$ ist ein Banach-, für $p=2$ ein Hilbertraum.