Exercise

Functional Analysis
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Papapetros Vaggelis
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Exercise

#1

Post by Papapetros Vaggelis »

Using the lemma http://www.mathimatikoi.org/forum/viewt ... =28&t=1133" onclick="window.open(this.href);return false;

prove that if \(\displaystyle{\left(H,\langle{,\rangle}\right)}\) is a Hilbert space and \(\displaystyle{U:H\to H}\)

is a \(\displaystyle{\mathbb{C}}\) -linear and bounded operator such that \(\displaystyle{||U||\leq 1}\),

with \(\displaystyle{F=\left\{h\in H\,,U(h)=h\right\}\,\,,N=\left\{U(h)-h\in H\,,h\in H\right\}}\) , then

\(\displaystyle{N=F^{\perp}}\).
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