# Disk algebra

In functional and complex analysis, the **disk algebra** *A*(**D**) (also spelled **disc algebra**) is the set of holomorphic functions

*f*:**D**→**C**,

where **D** is the open unit disk in the complex plane **C**, *f* extends to a continuous function on the closure of **D**. That is,

where *H*^{∞}(**D**) denotes the Banach space of bounded analytic functions on the unit disc **D** (i.e. a Hardy space).
When endowed with the pointwise addition (*f*+*g*)(*z*)=*f*(*z*)+*g*(*z*), and pointwise multiplication (*fg*)(*z*)=*f*(*z*)*g*(*z*), this set becomes an algebra over **C**, since if *ƒ* and *g* belong to the disk algebra then so do *ƒ* + *g* and *ƒg*.

Given the uniform norm,

by construction it becomes a uniform algebra and a commutative Banach algebra.

By construction the disc algebra is a closed subalgebra of the Hardy space *H*^{∞}. In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of *H*^{∞} can be radially extended to the circle almost everywhere.

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