replace double | with \Vert
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\begin{document}
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\newcommand{\K}{\mathrm{K}}
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\newcommand{\qnorm}[1]{\left|\left|#1\right|\right|^2}
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\newcommand{\norm}[1]{\left|\left|#1\right|\right|}
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\newcommand{\qnorm}[1]{\left\Vert#1\right\Vert^2}
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\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
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\section{Fourier-Entwicklung}
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\begin{definition}[Periodische Funktionen]
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$f: \R \to \K$ heißt $L$-periodisch $(L >0)$ falls $f(x+L) = f(x)$, $\forall x \in \R (\implies f(x+kL) = f(x),\; \forall k\in \Z)$.
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