Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality: For vectors $\mathbf{u}, \mathbf{v}$ in an inner product space: $$|\langle \mathbf{u}, \mathbf{v} \rangle|^2 \leq \langle \mathbf{u}, \mathbf{u} \rangle \cdot \langle \mathbf{v}, \mathbf{v} \rangle$$ For sums: $\left(\sum a_i b_i\right)^2 \leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)$. For expectations: $E[XY]^2 \leq E[X^2] \cdot E[Y^2]$. Equality holds iff $\mathbf{u}$ and $\mathbf{v}$ are proportional. When to use: Bounding sums or expectations that involve products. Proving that correlation $\leq 1$. Optimization over constrained sums. This is a fundamental technique used across probability, linear algebra, and analysis.