Prove that the norm on L^2 space is continuous

**Pur** · January 26th 2009, 05:39 PM

Prove that the norm, ${\left\lVert \cdot \right\rVert}$ , considered as a function ${\left\lVert \cdot \right\rVert}:\mathfrak{L}^2([-L,L],\mathbb{F})\rightarrow \mathbb{R}$ is continuous. i.e. $f_n\rightarrow f\in{\mathfrak{L}^2([-L,L],\mathbb{F})}$ , converges wrt the norm then, ${\left\lVert f_n \right\rVert}\rightarrow {\left\lVert f \right\rVert}\in{\mathbb{R}}$

**Opalg** · January 27th 2009, 12:16 AM

This is true in any normed space, and is a simple consequence of the triangle inequality. Start with the fact that $\|f_n\|\leqslant \|f\| + \|f_n-f\|$ , so that $\|f_n\| - \|f\| \leqslant\|f_n-f\|$ . The same reasoning with f and f_n interchanged then gives $\bigl|\|f_n\| - \|f\|\bigr| \leqslant\|f_n-f\|$ .

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