Real and Complex Analysis

A₁ assume Banach algebra Banach space Blaschke product Borel measure Borel set bounded linear functional compact set compact subset completes the proof complex function complex measurable complex numbers constant contains continuous function converges uniformly convex Corollary countable defined Definition dense differentiable disc disjoint entire function Exercise exists finite follows Fourier transform func function f ƒ and g ƒ ɛ harmonic function Hausdorff space Hence Hilbert space holds holomorphic functions implies inequality intersection L¹(T Lebesgue measure lemma mapping measurable function measure space o-algebra one-to-one open set plane Poisson integral polynomials positive measure PROOF Let properties real number region S-invariant satisfies semicontinuous shows simply connected subspace Suppose ƒ supremum Theorem Let Theorem Suppose tion u₁ union V₁ vector space zero