Editing Disk algebra

In mathematics, specifically in [[functional analysis|functional]] and [[complex analysis]], the '''disk algebra''' ''A''('''D''') (also spelled '''disc algebra''') is the set of [[holomorphic function]]s
: ''ƒ'' : '''D''' → <math>\mathbb{C}</math>
(where '''D''' is the [[open unit disk]] in the [[complex plane]] <math>\mathbb{C}</math>) that extend to a continuous function on the [[closure (topology)|closure]] of '''D'''.  That is,
: <math>A(\mathbf{D}) = H^\infty(\mathbf{D}) \cap C(\overline{\mathbf{D}}),</math>
where {{math|''H''<sup>&infin;</sup>('''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 {{nobr|(''f'' + ''g'')(''z'') {{=}} ''f''(''z'') + ''g''(''z'')}} and pointwise multiplication {{nobr|(''fg'')(''z'') {{=}} ''f''(''z'')''g''(''z''),}} this set becomes an [[algebra over a field|algebra]] over '''C''', since if ''f'' and ''g'' belong to the disk algebra, then so do ''f''&nbsp;+&nbsp;''g'' and&nbsp;''fg''.

Given the [[uniform norm]]
: <math>\|f\| = \sup\big\{|f(z)| \mid z \in \mathbf{D}\big\} = \max\big\{|f(z)| \mid z \in \overline{\mathbf{D}}\big\},</math>
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 infinity|''H''<sup>&infin;</sup>]]. In contrast to the stronger requirement that a continuous extension to the circle exists, it is [[Fatou's theorem|a lemma of Fatou]] that a general element of ''H''<sup>&infin;</sup> can be radially extended to the circle [[almost everywhere]].

== References ==
{{Reflist}}

{{Functional analysis}}
{{SpectralTheory}}

[[Category:Functional analysis]]
[[Category:Complex analysis]]
[[Category:Banach algebras]]

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