Hilbert space

A complete inner product space -- every Cauchy sequence in it converges to a limit still inside it.

A Hilbert space is an inner product space that is complete in the Cauchy-sequence sense: every sequence whose terms get arbitrarily close together converges to a limit that is still an element of the space. $L^2(-\pi,\pi)$ , the space this course's Fourier series live in, is a Hilbert space -- a fact that guarantees a Fourier series always converges to something in the same space, not just to something "outside" it.

Examples

$\mathbb R^n$ is a (finite-dimensional) Hilbert space; $L^2$ of square-integrable functions is the infinite-dimensional example this course cares about.