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Some of the many faces of the Cauchy-Schwarz inequality:

\[
|\langle u,v \rangle| \leq \|u\| \|v\|
\]
\[
\left| \sum_{i=1}^n u_i \bar{v}_i \right|^2 \leq \sum_{j=1}^n |u_j|^2 \sum_{k=1}^n |v_k|^2
\]
\(\left| \int_{\mathbb{R}^n} f(x)\overline{g(x)} dx \right|^2\)
\(\leq\int_{\mathbb{R}^n} |f(x)|^2 dx \int_{\mathbb{R}^n} |g(x)|^2 dx\)
\[
|E(XY)|^2 \leq E(X^2) E(Y^2)
\]
\[
\mathrm{Cov}(X,Y)^2 \leq \mathrm{Var}(X) \mathrm{Var}(Y)
\]
\[
|\varphi(g^* f)|^2 \leq \varphi(f^* f) \varphi(g^* g)
\]

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