Titlebook: Complex Analysis; A Functional Analysi D. H. Luecking,L. A. Rubel Textbook 1984 Springer-Verlag New York Inc. 1984 Analysis.Funktionalanaly

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,Runge’s Theorem,If f ∈ .(G), G a connected open set, it is a consequence of the power series expansion for holomorphic functions that if f(z.) = 0, z. → z. ∈ G then f = 0 in G. It is also a consequence that if f.(z.) = 0 for n = 0,1,2,…, then f = 0 in G. We adopt conventions about “sets with multiplicity” that allow us to treat both cases as one.

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The Riemann Mapping Theorem,The Riemann Mapping Theorem implies that, as far as . can tell, all simply connected regions are the “same”. To clarify what this means we need the following notion of equivalence.

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Dual Space Topologies,This chapter is intended as a prerequisite for later chapters. In it we introduce a topology on the dual of a topological vector space. We present some of the standard results in the theory of Fréchet spaces and some additional results on topological vector spaces in general.

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Preliminaries: Set Theory and Topology,tion of a family of sets, set difference (AB), complement (compl A), Cartesian product; functions, domain, range, one-to-one, onto, image, inverse, restriction; partial ordering, linear (or total) ordering, and equivalence relation.

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,Duality of ,(G)—The Case of the Unit Disc,of seminorms. For each non-empty finite set A = {‖•‖., ‖•‖.,…, ‖•‖.} ⊂ ., define., x ∈ E. Then ‖•‖. is a seminorm. Let . = . ∪ {‖•‖.: A is a non empty finite subset of .}; then . and . generate the same topology on E (Exercise 2). Consequently, we may assume . = . in the following proposition.