Cauchy-Schwarz inequality

In an inner product space over the field $R$ or $C$ , given the vectors $u$ and $u$

| ⟨ u, v ⟩ | \leq ∥ u ∥ \cdot ∥ v ∥

with equality holding $u$ and $v$ are linearly dependent.
Related: triangle inequality.

Examples:

In $C^{n}$ with the canonical complex inner product

{| u_{1} {\bar{v}}_{1} + \dots + u_{n} {\bar{v}}_{n} |}^{2} \leq ({| u_{1} |}^{2} + \dots + {| u_{n} |}^{2}) ({| v_{1} |}^{2} + \dots + {| v_{n} |}^{2})

For the inner product space $L^{2}$ of square-integrable complex-valued functions, the following inequality:

{| \int_{R^{n}} f (x) \overset{―}{g (x)} d x |}^{2} \leq \int_{R^{n}} | f (x) |^{2} d x \int_{R^{n}} | g (x) |^{2} d x .