---
tag: HM2HHE3J51
kind: concept
name: Hilbert space
msc: [46C05]
specializes: [06GZ956657, 0QVNCJXRY3]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert_space
same_as:
  wikidata: Q190056
  wikipedia: Hilbert space
---

## Summary

A Hilbert space is an [[inner product space|0QVNCJXRY3]] that is complete for the [[norm induced by its inner product|0MFJJV97RJ]].
So it is a [[Banach space|06GZ956657]] whose geometry has angles and orthogonality, as in Euclidean space.
Every Hilbert space has an orthonormal basis, and every closed subspace has an orthogonal complement.
Every continuous linear functional is the inner product with a fixed vector.
Hilbert spaces are the setting of Fourier series, of spectral theory and of quantum mechanics.

## Explanations

### L1 human

A Hilbert space is a space of vectors, possibly of infinite dimension, where lengths and angles work as in the plane.
The theorem of Pythagoras and orthogonal projections still make sense.
Completeness means that a sequence whose terms get closer and closer always has a limit in the space.
The main example is $\ell^2$, the sequences $(x_1, x_2, \dots)$ with $\sum \abs{x_n}^2$ finite.
Another is the space $L^2$ of square-integrable functions, where Fourier series live.