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Extensions of polynomials and holomorphic mappings arose in most of the previous chapters of this book, e.g. in estimating the polarization constant of the bidual (Corollary 1.52), in the definition of Q-reflexive Banach spaces (Definition 2.44), in studying the strong localization property for holomorphic germs (Proposition 4.37), in passing the relationship τ0 = τω to closed subspaces of fully nuclear spaces (Proposition 4.54) and, in Chapter 5, while proving the equality τ0 = τω on arbitrary open subsets of a Fréchet-Schwartz space with the bounded approximation property (Corollary 5.53) we used the holomorphic extension obtained in Lemma 5.38. In all these cases extensions were a means to an end. In this chapter we make them our primary concern. We have extended holomorphic functions from open sets to the envelope of holomorphy in Chapter 5. In the first two sections of this chapter we investigate:
extending holomorphic functions from E to F where E is a dense sub-space of F.
extending holomorphic functions from E to F where E is a closed sub-space of F.
KeywordsBanach Space Holomorphic Function Closed Subspace Convex Space Continuous Linear Operator
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© Springer-Verlag London Limited 1999