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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
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
In functional analysis, a uniform algebra ''A'' on a compact Hausdorff topological space ''X'' is a closed (with respect to the uniform norm) subalgebra of the C*-algebra ''C(X)'' (the continuous complex-valued functions on ''X'') with the followi ...
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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