Complex Analysis

1. Front Matter

   Pages i-vii

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2. Preliminaries: Set Theory and Topology

   • D. H. Luecking, L. A. Rubel

   Pages 1-12

3. Preliminaries: Vector Spaces and Complex Variables

   • D. H. Luecking, L. A. Rubel

   Pages 13-26

4. Properties of C(G) and H(G)

   • D. H. Luecking, L. A. Rubel

   Pages 27-32

5. More about C(G) and H(G)

   • D. H. Luecking, L. A. Rubel

   Pages 33-37

6. Duality

   • D. H. Luecking, L. A. Rubel

   Pages 38-43

7. Duality of H(G)—The Case of the Unit Disc

   • D. H. Luecking, L. A. Rubel

   Pages 44-50

8. The Hahn-Banach Theorem, and Applications

   • D. H. Luecking, L. A. Rubel

   Pages 51-59

9. More Applications

   • D. H. Luecking, L. A. Rubel

   Pages 60-66

10. The Dual of H(G)

    • D. H. Luecking, L. A. Rubel

    Pages 67-76

11. Runge’s Theorem

    • D. H. Luecking, L. A. Rubel

    Pages 77-83

12. The Cauchy Theorem

    • D. H. Luecking, L. A. Rubel

    Pages 84-95

13. Constructive Function Theory

    • D. H. Luecking, L. A. Rubel

    Pages 96-107

14. Ideals in H(G)

    • D. H. Luecking, L. A. Rubel

    Pages 108-116

15. The Riemann Mapping Theorem

    • D. H. Luecking, L. A. Rubel

    Pages 117-123

16. Carathéodory Kernels and Farrell’s Theorem

    • D. H. Luecking, L. A. Rubel

    Pages 124-129

17. Ring (not Algebra) Isomorphisms of H(G)

    • D. H. Luecking, L. A. Rubel

    Pages 130-135

18. Dual Space Topologies

    • D. H. Luecking, L. A. Rubel

    Pages 136-150

19. Interpolation

    • D. H. Luecking, L. A. Rubel

    Pages 151-156

20. Gap-Interpolation Theorems

    • D. H. Luecking, L. A. Rubel

    Pages 157-167

The main idea of this book is to present a good portion of the standard material on functions of a complex variable, as well as some new material, from the point of view of functional analysis. The main object of study is the algebra H(G) of all holomorphic functions on the open set G, with the topology on H(G) of uniform convergence on compact subsets of G. From this point of vie~, the main theorem of the theory is Theorem 9.5, which concretely identifies the dual of H(G) with the space of germs of holomorphic functions on the complement of G. From this result, for example, Runge's approximation theorem and the global Cauchy integral theorem follow in a few short steps. Other consequences of this duality theorem are the Germay interpolation theorem and the Mittag-Leffler Theorem. The approach via duality is entirely consistent with Cauchy's approach to complex variables, since curvilinear integrals are typical examples of linear functionals. The prerequisite for the book is a one-semester course in com­ plex variables at the undergraduate-graduate level, so that the elements of the local theory are supposed known. In particular, the Cauchy Theorem for the square and the circle are assumed, but not the global Cauchy Theorem in any of its forms. The second author has three times taught a graduate course based on this material at the University of Illinois, with good results.

• Analysis
• Funktionalanalysis
• Funktionentheorie
• calculus
• functional analysis

• Department of Mathematics, University of Arkansas, Fayetteville, USA

  D. H. Luecking

• Department of Mathematics, University of Illinois, Urbana-Champaign, USA

  L. A. Rubel

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