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Let $H$ be a Hilbert space and let y be an element of $H-{0}$. Let $S$ be the linear span of $y$.

How can we show that the orthogonal complement of $\{x\in H:\langle x,y\rangle=0\}$ is $S$?

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Hint: Any vector $x$ can be written in the form $x=\lambda y+z$ for some $\lambda\in\Bbb C$ and $z\perp y$: to see this, you can find $\lambda$ by inner multiplication with $y$.

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