Proposition 12.6.6 (Cauchy-Schwarz Inequality).label Let $H$ be a vector space over $K \in \RC$ and $\inp_{H}: E \times E \to K$ be an inner product, then for any $x, y \in H$, $\dpn{x, y}{H}\le \norm{x}_{H}\norm{y}_{H}$.
Proof, [Theorem 5.19, Fol99]. Assume without loss of generality that $\dpn{x, y}{H}> 0$, then for each $t \in \real$,
\[0 \le \dpn{x - ty,x - ty}{H}= \norm{x}_{H}^{2} - 2t\dpn{x, y}{H}+ t^{2}\norm{y}_{H}^{2}\]
which is a quadratic function of $t$ with absolute minumum at $t = \dpn{x, y}{H}/\norm{y}_{H}^{2}$, so
\[0 \le \norm{x}_{H}^{2} - 2\dpn{x, y}{H}^{2}/\norm{y}_{H}^{2} + \dpn{x, y}{H}^{2}/\norm{y}_{H}^{2} = \norm{x}_{H}^{2} - \dpn{x, y}{H}^{2}/\norm{y}_{H}^{2}\]
$\square$
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