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Notation and Terminology
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Banach space bounded on SC bounded subset Cauchy inequalities Cauchy-Hadamard formula closed ball compact subset compact-open topology conditions is necessary containing current holomorphy type defined denote dff(x dmf(x easily seen emsup equicontinuous equilibrated Ergebnisse der Mathematik fact find a real finite dimensional following lemma Hence holomorphic functions Holomorphic Mappings induced Jmf(x Let feJf(U;F Nachbin natural topology normed vector space number e>0 open subset Pm(x ported by compact power series Proof of Proposition prove Q. E. D. Corollary Q. E. D. Proposition Q. E. D. Remark Q.E.D. Proof radius of convergence real number c(V)>0 real numbers C^0 relatively compact S C is relatively SC is bounded seminorm sense sequence series of f spaces of holomorphic subset S C Taylor series theorem topologies J0 topology on Jf(U;F Topology on Spaces vector subspace XxeU