---
tag: 528G11SDWR
kind: concept
name: Schauder basis
msc: [46B15]
sources:
  - url: https://en.wikipedia.org/wiki/Schauder_basis
same_as:
  wikidata: Q777879
  wikipedia: Schauder basis
---

## Summary

A Schauder basis of a [[Banach space|06GZ956657]] is a sequence $(e_n)$ such that every vector is the sum of a unique norm-convergent series $\sum \alpha_n e_n$.
It replaces the finite linear combinations of a [[basis|0KCQ9Y3QKA]] by infinite series, and so it fits infinite-dimensional spaces.
The unit vectors form a Schauder basis of $\ell^p$ for $1 \le p < \infty$ and of $c_0$.
A Banach space with a Schauder basis is separable, but Enflo showed that not every separable Banach space has a Schauder basis.

## Explanations

### L1 human

In $\R^n$ every vector is a combination of the $n$ unit vectors.
In a space of infinite sequences one would like to write every vector as an infinite combination $\alpha_1 e_1 + \alpha_2 e_2 + \cdots$ of basic vectors.
A Schauder basis is a sequence of vectors for which this works, with the coefficients determined uniquely and the series converging in the norm of the space.
For example, a sequence $x$ in $\ell^2$ is the sum of the series $x_1 e_1 + x_2 e_2 + \cdots$, where $e_k$ has $1$ in place $k$ and $0$ elsewhere.