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Consider $$T: (C[-1,1],\|\cdot\|_{2})\rightarrow \mathbb{C}\\Tf :=\int_{-1}^{1}mf\,\mathrm{d}x$$ where $m\in C[-1,1]$.

I want to prove $\|T\| = \|m\|_2$.

$\|T\|\leq\|m\|_2$ can be easily proved by Hölder's inequality, how to solve $\|T\|\geq\|m\|_{2}$?

Thank you.

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Find an $f$ with $\|f\|_2=1$ such that $Tf=\|m\|_2$? – Christian Clason Nov 28 '12 at 18:44
up vote 3 down vote accepted

Use $m/\|m\|_2$.$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $

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