In function theory, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s

f : D → C,

where D is the open unit disk in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 C, f extends to a continuous function on the closure

Closure (topology)

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

 of D. That is,
where denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a Hardy space

Hardy space

In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...

).
When endowed with the pointwise addition,
(f+g)(z)=f(z)+g(z),
and pointwise multiplication,(z)=f(z)g(z),

this set becomes an algebra

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

 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

Uniform algebra

A uniform algebra A on a compact Hausdorff topological space X is a closed subalgebra of the C*-algebra C with the following properties:As a closed subalgebra of the commutative Banach algebra C a uniform algebra is itself a unital commutative Banach algebra A uniform algebra A on a compact...

 and a commutative Banach algebra

Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...

. By construction it is a subalgebra of the Hardy space

Hardy space

In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...

 H^∞.