---
tag: ZYDYD7FB1Z
kind: concept
name: Riesz basis
msc: [42C15, 46B15, 46C05]
specializes: [528G11SDWR]
sources:
  - url: https://en.wikipedia.org/wiki/Riesz_sequence
---

## Summary

A Riesz basis of a [[Hilbert space|HM2HHE3J51]] is the image of an [[orthonormal basis|0S9GGTR0EX]] under a bounded invertible operator.
It keeps the main features of an orthonormal basis.
Every vector has a unique expansion, and its length is comparable to the length of its coefficient sequence.
Riesz bases are exactly the bases that are also frames, and they are unconditional [[Schauder bases|528G11SDWR]].
They are the natural notion of a stable basis in sampling theory, wavelets and nonharmonic Fourier analysis.

## Explanations

### L1 human

An orthonormal basis is like a set of perpendicular rulers of length one.
A Riesz basis is what you get after you stretch and tilt the space by a linear map that can be undone.
The rulers are no longer perpendicular or of equal length, but they are not too close to each other either.
So coordinates in a Riesz basis are still unique and still control the length of the vector, up to fixed constants.