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Analytic Functions application Baire Category Theorem Banach space belongs boundary bounded inverse bounded linear functional Closed Graph Theorem closed linear operator closed linear subspace closure compact operator compact set complete Consider constant contains a sphere continuous inverse continuous linear functional continuous map converges convex set D.S. Chapter denote dense set differential equation disjoint domain element finite dimensional finite number fixed point follows functional f Furthermore graph Hahn-Banach Theorem Hence Hilbert space implicit function theorem inequality interior point internal point Lemma linear functional defined linear normed space linear subspace linear topological space metric space neighborhood normed linear space null space open set origin Problem Proof of Theorem prove reflexive Remark Riesz satisfying scalar sequence sp(T spaoe spectrum subsequence Suppose three lines theorem uniformly bounded uniformly convex unit sphere unit vector valued functions vanishes whioh XI-T zero