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In functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions f : D → C, where D is the open unit disk in the complex plane C, f extends to a continuous function on the closure of D. That is, where H∞(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 (f+g)(z)=f(z)+g(z), and pointwise multiplication (fg)(z)=f(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,

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  • In functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions f : D → C, where D is the open unit disk in the complex plane C, f extends to a continuous function on the closure of D. That is, where H∞(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 (f+g)(z)=f(z)+g(z), and pointwise multiplication (fg)(z)=f(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,
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  • Disk algebra
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  • In functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions f : D → C, where D is the open unit disk in the complex plane C, f extends to a continuous function on the closure of D. That is, where H∞(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 (f+g)(z)=f(z)+g(z), and pointwise multiplication (fg)(z)=f(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 H∞. 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.
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