Cauchy-Schwarz inequality
In an inner product space over the field $\mathbb{R}$ or $\mathbb{C}$, given the vectors $u$ and $u$
$$
|\langle u,v \rangle| \leq \| u \|\cdot\| v \|
$$
with equality holding $u$ and $v$ are linearly dependent.
Examples:
- In $\mathbb{C}^n$ with the canonical complex inner product
$$
\left|u_{1} \bar{v}_{1}+\cdots+u_{n} \bar{v}_{n}\right|^{2} \leq\left(\left|u_{1}\right|^{2}+\cdots+\left|u_{n}\right|^{2}\right)\left(\left|v_{1}\right|^{2}+\cdots+\left|v_{n}\right|^{2}\right)
$$
- For the inner product space $L^2$ of [square-integrable](https://en.wikipedia.org/wiki/Square-integrable "Square-integrable") complex-valued functions, the following inequality:
$$
\left|\int_{\mathbb{R}^{n}} f(x) \overline{g(x)} d x\right|^{2} \leq \int_{\mathbb{R}^{n}}|f(x)|^{2} d x \int_{\mathbb{R}^{n}}|g(x)|^{2} d x .
$$
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Author of the notes: Antonio J. Pan-Collantes
antonio.pan@uca.es
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