- f : D → C,
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, (f+g)(z)=f(z)+g(z), and pointwise multiplication,
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 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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