---
tag: DGWYF0B6NT
kind: concept
name: Disk algebra
msc: [46J15, 30H50]
sources:
  - url: https://en.wikipedia.org/wiki/Disk_algebra
same_as:
  wikidata: Q2676720
  wikipedia: Disk algebra
---

## Summary

The disk algebra $A(\mathbb{D})$ is the algebra of continuous functions on the closed unit disc that are holomorphic inside it.
With the supremum norm it is a commutative Banach algebra, and the polynomials are dense in it.
It is the basic example of a uniform algebra and a natural setting for boundary problems on the [[unit disc|8G5ABXRK85]].

## Explanations

### L1 human

Take a function on the closed disc that is continuous up to the edge and holomorphic in the interior.
Such functions can be added and multiplied, and uniform limits of them have the same properties.
Polynomials are examples, and every element is a uniform limit of polynomials.
The function $\sum_{n \ge 1} z^n / n^2$ is in the disk algebra, since the series converges uniformly on the closed disc.
The function $1/(1 - z)$ is holomorphic in the open disc but does not belong, because it is unbounded near $1$.