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AA(U AH(X algebra of holomorphic analytic function algebra approximation Bishop's Theorem Cauchy filter Chapter choose compact holomorphic compact holomorphicly convex compact neighborhood compact set compact subset complex analytic manifold component contradiction convergence on compact Corollary countable denote differentiably stable analytic domain of holomorphy exists f e f e Hol(W functions f fundamental system Hence HoKU Hol(M Hol(U Hol(V holomorphic functions holomorphic set holomorphicly convex set homomorphism space inductive limit topology intersection Lemma manifold spread Mer(X meromorphic one-one open neighborhood open set open subset Poly Poly-hull(X polydisc polyhedra polynomial F proof of Theorem proper map properties Rat-hull(X rational relatively compact seminorm Serre duality set containing special holomorphic polyhedron stable analytic algebra stable uniform algebra Stein manifold submanifold subset of ft subvariety Suppose system of neighborhoods uniform convergence unit polynomial weak peak point zero dimensional