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Let $\mathcal{H}$ be a Hilbert space. By the Riesz Representation Theorem, we have that any bounded linear $\psi \in \mathcal{H}^{*}$ is of the form $\psi(h) = \langle h, g \rangle$ for some $g \in \mathcal{H}$. I need to determine $g$ such that $\|\psi\| = 1$ and $\psi(h) = \|h\|$. I tried replacing $g = \displaystyle \frac{h}{\|h\|}$ but not luck. Any help will be deeply appreciated.

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1 Answer 1

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You did good.

If $\phi=(\cdot,g)$ then $\|\phi\|=\|g\|$, i.e. Riesz representation is an isometric isomorphism. Indeed, by Cauchy-Schwarz, $\|\phi\|\leq\|g\|$. And $\phi(g)=\|g\|^2$ so $\phi(g/\|g\|)=\|g\|$ hence $\|\phi\|\geq \|g\|$.

Now you see that $g=\frac{h}{\|h\|}$ yields $\|\phi\|=\|g\|=1$ and $\phi(h)=(h,g)=\|h\|^2/\|h\|=\|h\|$.

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Thank you very much for the help! –  user44069 Mar 12 '13 at 4:18
    
@Stefan You're welcome. Glad it helped. –  1015 Mar 12 '13 at 4:19

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