# arrow_back What is the Cauchy-Schwartz inequality useful for in mathematics?

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What is the Cauchy-Schwartz inequality useful for in mathematics?

This inequality is sometimes useful in proving bounds:
$$|\langle\mathbf{x}, \mathbf{y}\rangle| \leq\|\mathbf{x}\| \cdot\|\mathbf{y}\|$$
for all $\mathrm{x}, \mathrm{y} \in V$. Equality holds exactly when $\mathrm{x}$ and $\mathrm{y}$ are scalar multiples of each other (or equivalently, when they are linearly dependent).
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