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ASPECTS OF HOLOMORPHIC bornological bounded sets bounded subset C U and f Cauchy inequalities Center for Theoretical compact open topology compact set K C U compact subset compact-open compact-open topology complex Banach spaces Continuous Multilinear Mappings convex topological vector Coral Gables cP(mE;F define a topology Definition denote dimensional and F dimensional or F=0 dm f easy to show equicontinuity F be complex f c is relatively f c M(U;F f is bounded family of seminorms finite dimensional HOLOMORPHIC MAPPINGS Leopoldo inf(fic,d infinite Lemma lim sup locally convex topological MAPPINGS Leopoldo Nachbin maps are continuous mE;F mE:F metrizable norm onC(U;F open set P-m sup polarization formula positive number power series Proof Proposition Pura e Aplicada relatively compact Remark series of dmf series of f symmetric mappings Taylor series Theorem Theoretical Studies University topological vector space uniform convergence University of Miami vector subspace York and Instituto