In mathematics, specifically in
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
and
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the disk algebra ''A''(D) (also spelled disc algebra) is the set of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s
:''ƒ'' : D →
,
(where D is the
open unit disk in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
) that extend to a continuous function on the
closure of D. That is,
:
where denotes the Banach space of bounded analytic functions on the unit disc D (i.e. a
Hardy space).
When endowed with the pointwise addition (''ƒ'' + ''g'')(''z'') ''ƒ''(''z'') + ''g''(''z''), and pointwise multiplication (''ƒg'')(''z'') ''ƒ''(''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 . 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.
References
Functional analysis
Complex analysis
Banach algebras
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