Disk algebra
In mathematics, specifically in functional and complex analysis, the disk algebra "A"(D) (also spelled disc algebra) is the set of holomorphic functions (where D is the open unit disk in the complex plane ℂ) 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.