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Let $(V, \langle \hspace{1mm} , \hspace{1mm} \rangle)$ be an inner product space, $x \in V$ a unit vector, and $y \in V.$ Prove $$\langle y,x \rangle \langle x,y \rangle \leq \langle y,y \rangle.$$

How do we prove this inequality? I've tried using the Cauchy-Schwarz inequality $|\langle x,y \rangle| \leq \|x\| \|y \|$ combined with the fact that $\|x\|=1$ but I am having trouble. Any help appreciated

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I do not see where you might be having problem. Using your approach, it simply follows that $$\langle x,y\rangle\langle y,x\rangle=|\langle x,y\rangle|^2\le \|x\|^2\|y\|^2=\|y\|^2=\langle y,y\rangle$$ by Cauchy-Schwarz.

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