Inner product space

A vector space equipped with an inner product, generalizing the dot product.

A vector space $V$ becomes an inner product space once it carries an inner product -- an operation generalizing the dot product that assigns a number to every pair of vectors and satisfies the same handful of axioms (linearity, symmetry, positivity). The space of piecewise-continuous functions on an interval, with $\langle f,g\rangle=\int_a^b f(x)g(x)\,dx$ , is the running example this course builds its entire geometric picture of Fourier series from.

Examples

$\mathbb R^n$ with the ordinary dot product is the simplest inner product space; $L^2(-\pi,\pi)$ with the integral inner product is another.