## Introduction to HolomorphyThis book presents a set of basic properties of holomorphic mappings between complex normed spaces and between complex locally convex spaces. These properties have already achieved an almost definitive form and should be known to all those interested in the study of infinite dimensional Holomorphy and its applications. The author also makes ``incursions'' into the study of the topological properties of the spaces of holomorphic mappings between spaces of infinite dimension. An attempt is then made to show some of the several topologies that can naturally be considered in these spaces. Infinite dimensional Holomorphy appears as a theory rich in fascinating problems and rich in applications to other branches of Mathematics and Mathematical Physics. |

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amply bounded as(mE;F Banach space bounded subset bounded type Cauchy inequalities Cauchy integral formula compact set compact subset compact-open topology complex locally convex connected open subset continuous converges uniformly CS(E CS(F defined DEFINITION denote F be separated f is holomorphic F is separated finite dimensional finitely holomorphic following are equivalent g E U hence holomorphic mapping identically zero implies Infinite Dimensional Holomorphy intentionally left blank Lemma linear locally bounded locally convex spaces locally convex topology lsism m-homogeneous polynomial m-linear mappings mapping f mE;F Nachbin natural topology non-empty connected open non-empty open subset normed space North-Holland Pub Pa(mE;F Pm(x polarization formula ported power series precompact prove Q.E.D. PROPOSITION Q.E.D. REMARK radius of convergence seminorm sequence series of f sup Hf(X)H Suppose Taylor series theorem U-bounded ub(U vector space weak topology X E C