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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.

Inhaltsverzeichnis

Frontmatter

§ 1. Preliminaries: Set Theory and Topology

Abstract
We assume familiarity with the rudiments of informal set theory including such notions as set, subset, superset, the null set Ɉ, the union or intersection of a family of sets, set difference (A\B), complement (compl A), Cartesian product; functions, domain, range, one-to-one, onto, image, inverse, restriction; partial ordering, linear (or total) ordering, and equivalence relation.
D. H. Luecking, L. A. Rubel

§ 2. Preliminaries: Vector Spaces and Complex Variables

Abstract
Our goal is the treatment of complex analysis from the point of view of topological vector spaces. Here we present the basic definitions and some well-known facts.
D. H. Luecking, L. A. Rubel

§ 3. Properties of C(G) and H(G)

Abstract
With little change we can study functions of several variables. To keep the notation simple, we will restrict ourselves to two variables. In that case, G is an open set in ℂ × ℂ = ℂ2. Then C(G) is defined just as in the one-variable case. (The distance in ℂ2 between two points (z,w) and (z’,w’) will be denoted d((z,w), (z’,w’)) = (|z - z’|2 + |w - w’|2)½. We also define H(G) as before, but must first define holomorphic.
D. H. Luecking, L. A. Rubel

§ 4. More about C(G) and H(G)

Abstract
It is commonly prove in courses of advanced calculus that compact sets in \( \mathbb{R} \) (or more generally \( {\mathbb{R}^n} \)) are characterized by being closed and bounded. In a general topological vector space only one implication is correct. Here and in the future we will assume that our topological vector spaces are Hausdorff.
D. H. Luecking, L. A. Rubel

§ 5. Duality

Abstract
The concept of duality is one of the most productive in analysis. One can very nearly characterize applications of functional analysis as applications of duality. Most of our goal in succeeding sections will be identifying the dual of H(G) and exploiting that identification. Toward this end, we begin with the definition.
D. H. Luecking, L. A. Rubel

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

Abstract
We begin with a general result about linear functionals on a locally convex topological vector space. Let E have the topology generated by a family P of seminorms. For each non-empty finite set A = {‖•‖1, ‖•‖2,…, ‖•‖n} ⊂ P, define\( {\left\| x \right\|_A} = \mathop{{\max }}\limits_{{l \leqslant j \leqslant n}} \,{\left\| x \right\|_j} \), x ∈ E. Then ‖•‖A is a seminorm. Let = P ∪ {‖•‖A: A is a non empty finite subset of P}; then P and generate the same topology on E (Exercise 2). Consequently, we may assume P = in the following proposition.
D. H. Luecking, L. A. Rubel

§ 7. The Hahn-Banach Theorem, and Applications

Abstract
A major tool in the application of duality results (in any locally convex topological vector space) is the Hahn-Banach Theorem. We state here one standard version (there are many equivalent versions) and two important corollaries.
D. H. Luecking, L. A. Rubel

§ 8. More Applications

Abstract
In this section E will denote the space H(ℂ), i.e., the space of entire functions with the topology of uniform convergence on compact sets.
D. H. Luecking, L. A. Rubel

§ 9. The Dual of H(G)

Abstract
We want to prove, as in the case of the disk, that H(G)* = H 0(ℂ \ G). We first study the dual of C(G). We change our notation here and write L(f) = ∫ fdμ when L ∈ C(G)*. (For the reader unfamiliar with integration theory this is simply a change in notation: The left-hand side defines the right-hand side. There are two advantages to this notation. First, it is the notation in which research papers are written. Second, the reader can call upon her experience with integration for intuition. For the mathematically advanced reader: we are invoking the Riesz Representation Theorem for C(G) * .) We call μ the “measure” associated with L, and we may identify μ and L. The collection of all such μ is denoted M0(G), so that M0(G) = C(G)*. We also write L(f) = ∫ f(z)dμ(z) when it is necessary to indicate the independent variable. “Measures” have the same properties as continuous linear functionals (which is what they are); for reinforcement, we list them here. Given μ ∈ M0(G):
i)
∫ (f + g)dμ = ∫ fdμ + ∫ gdμ, f, g ∈ C(G).
 
ii)
∫ afdμ = a ∫ fdμ, f ∈ C(G), a ∈ ℂ.
 
iii)
If fn → f in C(G) then ∫ fndμ → ∫ fdμ.
 
iv)
There is a compact set K ⊆ G such that | ∫ fdμ | ≤ C‖f‖K for all f ∈ C(G).
 
D. H. Luecking, L. A. Rubel

§ 10. Runge’s Theorem

Abstract
If f ∈ H(G), G a connected open set, it is a consequence of the power series expansion for holomorphic functions that if f(zn) = 0, zn → z0 ∈ G then f = 0 in G. It is also a consequence that if f(n)(z0) = 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.
D. H. Luecking, L. A. Rubel

§ 11. The Cauchy Theorem

Abstract
Runge’s Theorem can be used to prove Cauchy’s Theorem. This will require the elements of integration theory described in Chapter 2. Recall that for a rectifiable curve γ: [0,1] → ℂ, we let ‖γ‖ denote its length, i.e. ‖γ‖ = <Inline>1</Inline> |γ′(t)|dt, so that
$$ \left| {\int_{\gamma } {f(z)dz} } \right| \leqslant \left\| \gamma \right\| \cdot \left\| f \right\| $$
where ‖f‖ = sup{| f(z) |: z ∈ γ}. The reader is reminded that γ^ denotes the “physical curve”, that is the image of γ.
D. H. Luecking, L. A. Rubel

§ 12. Constructive Function Theory

Abstract
The goal of this section is the construction, by means of sums and products of simpler functions, of holomorphic functions with prescribed behavior. In what follows, when we speak of a sequence of complex numbers we ordinarily mean a sequence with multiplicity, so that a function taking some value at a point of the sequence must take that value with the appropriate multiplicity. We also exclude the trivial function f ≡ 0 unless otherwise noted.
D. H. Luecking, L. A. Rubel

§ 13. Ideals in H(G)

Abstract
We use the results of the previous section to derive some descriptive results on ideals of holomorphic functions.
D. H. Luecking, L. A. Rubel

§ 14. The Riemann Mapping Theorem

Abstract
The Riemann Mapping Theorem implies that, as far as H(G) can tell, all simply connected regions are the “same”. To clarify what this means we need the following notion of equivalence.
D. H. Luecking, L. A. Rubel

§ 15. Carathéodory Kernels and Farrell’s Theorem

Abstract
Given a sequence {Gn} of regions and a region G ≠ Ɉ such that G ⊆ Gn+1 ⊆ Gn for n = 1,2,3,.., we say that a superset G’ of G is suitable if G’ is connected and G’ ⊆ ∩Gn. Then ker[Gn: G], the kernel of {Gn} with respect to G, is defined as the union of all suitable supersets of G.
D. H. Luecking, L. A. Rubel

§ 16. Ring (not Algebra) Isomorphisms of H(G)

Abstract
We return here to the ring structure of H(G). A ring homomorphism of R1 to R2 is a function φ: R1 → R2 which preserves multiplication i.e. φ(rs) = φ(r)φ(s) and φ(r + s) = φ(r) + φ(s) for all r, s ∈ R1. A ring isomorphism is a ring homomorphism that is one-to-one and onto. If G and G’ are two conformally equivalent domains in ℂ then there is an algebra isomorphism from H(G) to H(G’) as we saw in Chapter 5. An algebra isomorphism will be a ring isomorphism which additionally preserves scalar multiplication. It follows from Proposition 5.4 that H(G) and H(G’) are isomorphic as algebras if and only if G and G’ are conformally equivalent. But a ring isomorphism can exist without conformal equivalence.
D. H. Luecking, L. A. Rubel

§ 17. Dual Space Topologies

Abstract
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.
D. H. Luecking, L. A. Rubel

§ 18. Interpolation

Abstract
In this chapter we study we study interpolation by holomorphic functions from a different point of view from that used in 12. In particular, the Germay Theorem (12.14) says that there exists an entire function that interpolates given values at a given sequence of points. There are three major approaches that can be taken to proving such a theorem. The first is via the Mittag-Leffler Theorem and Weierstrass products, which involves writing an explicit formula. The second is via solving infinitely many linear equations in infinitely many unknowns, the Taylor coefficients. (See [M. Eidelheit] and [P. J. Davis].) The third is via functional analysis—specifically the Banach-Dieudonné theorem. Kere we take the third route, obtaining in the process a functional analysis proof of Theorem 12.18.
D. H. Luecking, L. A. Rubel

§ 19. Gap-Interpolation Theorems

Abstract
There are many theorems in classical analysis where gaps play a rôle. We take up now some considerations from [N. Kalton and L. A. Rubel] where gaps and interpolation are mixed. The idea is to take the Germay interpolation situation, where we want f(zn) = wn, n = 1,2,3,… for some entire function f but now require that f have the form
$$ f(z) = \mathop{\Sigma }\limits_{{\lambda \in \Lambda }} \,{a_{\lambda }}{z^{\lambda }} $$
where ⋀ is a given set of positive integers. For certain ⋀ (like ⋀ = \( \Lambda = \mathbb{N} \), the set of all positive integers), this interpolation is always possible—provided we require |zn| → ∞ (and no zn = 0).
D. H. Luecking, L. A. Rubel

§ 20. First-Order Conformal Invariants

Abstract
The Theme of this section is the following: Suppose you find yourself on a plane domain, with only a restricted logic at your disposal; how closely can you determine which domain you are on—up to conformal equivalence? This leads to a study of a system of conformal invariants, the first-order conformal invariants (FOCI), which are obtained from the elementary properties of the algebra (or ring) of analytic functions on plane domains. Although the formal definition of FOCI is given in the terminology of mathematical logic, these invariants are nonetheless all included within the framework of classical function theory. Each of the FOCI corresponds to an elementary assertion about analytic functions that can be understood without any knowledge of mathematical logic.
D. H. Luecking, L. A. Rubel

Backmatter

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