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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.

  • 0
    Find an $f$ with $\|f\|_2=1$ such that $Tf=\|m\|_2$?2012-11-28

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