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Thread: Prove that the norm on L^2 space is continuous

  1. #1
    Pur
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    Prove that the norm on L^2 space is continuous

    Prove that the norm,$\displaystyle {\left\lVert \cdot \right\rVert}$, considered as a function $\displaystyle {\left\lVert \cdot \right\rVert}:\mathfrak{L}^2([-L,L],\mathbb{F})\rightarrow \mathbb{R}$ is continuous. i.e. $\displaystyle f_n\rightarrow f\in{\mathfrak{L}^2([-L,L],\mathbb{F})}$, converges wrt the norm then, $\displaystyle {\left\lVert f_n \right\rVert}\rightarrow {\left\lVert f \right\rVert}\in{\mathbb{R}}$
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    This is true in any normed space, and is a simple consequence of the triangle inequality. Start with the fact that $\displaystyle \|f_n\|\leqslant \|f\| + \|f_n-f\|$, so that $\displaystyle \|f_n\| - \|f\| \leqslant\|f_n-f\|$. The same reasoning with f and f_n interchanged then gives $\displaystyle \bigl|\|f_n\| - \|f\|\bigr| \leqslant\|f_n-f\|$.
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