---
tag: 40SV748R64
kind: concept
name: Hilbert dimension
msc: [46C05]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert_space
---

## Summary

The Hilbert dimension of a [[Hilbert space|HM2HHE3J51]] is the cardinality of any of its orthonormal bases, that is of any maximal [[orthonormal family|0RW9TNNT87]].
All orthonormal bases of a given Hilbert space have the same cardinality, so this number is well defined.
Two Hilbert spaces over the same field are isometrically isomorphic exactly when they have the same Hilbert dimension.
A Hilbert space is separable exactly when its Hilbert dimension is finite or countable.

## Explanations

### L1 human

The Hilbert dimension counts the perpendicular directions of unit length needed to build every vector of the space as a convergent series.
For a finite-dimensional space it is the usual dimension.
For $\ell^2$ it is countably infinite, with the unit vectors $e_1, e_2, \dots$ as basis.
This count differs from the algebraic dimension, which only allows finite sums and is uncountable for $\ell^2$.