Applications of Model Theory to Complex Analysis Citation Stroyan, Keith Duncan (1971) Applications of Model Theory to Complex Analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/69R3-SY38. https://resolver.caltech.edu/CaltechTHESIS:04112018-095018367 Abstract We use a nonstandard model of analysis to study two main topics in complex analysis. UNIFORM CONTINUITY AND RATES OF GROWTH OF MEROMORPHIC FUNCTIONS is a unified nonstandard approach to several theories; the Julia-Milloux theorem and Julia exceptional functions, Yosida's class (A), normal meromorphic functions, and Gavrilov's W p classes. All of these theories are reduced to the study of uniform continuity in an appropriate metric by means of S-continuity in the nonstandard model (which was introduced by A. Robinson). The connection with the classical Picard theorem is made through a generalization of a result of A. Robinson on S-continuous *-holomorphic functions. S-continuity offers considerable simplifications over the standard sequential approach and permits a new characterization of these growth requirements. BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A BANACH SPACE is a nonstandard approach to the pre-dual Banach space for H ∞ (D) which was introduced by Rubel and Shields. A new characterization of the pre-dual by means of the nonstandard hull of a space of contour integrals infinitesimally near the boundary of an arbitrary region is given. A new characterization of the strict topology is given in terms of the infinitesimal relation: "h b k provided ||h-k|| is finite and h(z) ≈ k(z) for z∈(*D)". A new proof of the noncoincidence of the strict and Mackey topologies is given in the case of a smooth finitely connected region. The idea of the proof is that the infinitesimal relation: "h γ k provided ||h-k|| is finite and h(z) ≈ k(z) on nearly all of the boundary", gives rise to a compatible topology finer than the strict topology. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Bohnenblust, Henri Frederic (advisor) Luxemburg, W. A. J. (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1971 Funders: Funding Agency Grant Number NSF UNSPECIFIED National Defense Education Act (NDEA) UNSPECIFIED Ford Foundation UNSPECIFIED Record Number: CaltechTHESIS:04112018-095018367 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04112018-095018367 DOI: 10.7907/69R3-SY38 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10798 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 11 Apr 2018 20:12 Last Modified: 27 Jun 2024 17:18 Thesis Files Preview PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

APPLICATIONS OF MODEL THEORY TO COMPLEX ANALYSIS Thesis by Keith Duncan Stroyan In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1971 ii ACK.NOWLEffiEME:NT I acknowledge with hearty thanks the guidance I have received f r ot1 Professors H. F. Bol:1lJ.. enblust and W. A. J. Luxemburg. I especially appreciate having been able to attend last summer's Oberwolfach symposiwn, which Profes s or Luxemburg made possible, and the many suggestions he made for the paper I presented there. The teaching experiences gained under the supervision of Professor R. A. Dean are also appreciated. I am grateful for the financial aid of the National Science Foundation, NDEA, and the Ford Foundation. A final thanks goes to all those who helped me find a job in this year's difficult market. ABSTRACT OF iii APPLICATIONS OF MODEL THEORY TO COMPLEX ANALYSIS by Keith Duncan Stroyan We use a nonstandard model of analysis to study two main topics in complex analysis. UNIFORM CONTIN'UITY AND RATES OF GROWTH OF MEROMORPHIC FUNG TIONS is a unified nonstandard approach to several theories; the Julia-Milloux theorem and Julia exceptional functions, Yosida's class (A), normal meromorphic functions, and Gavrilov's W p classes. All of these theories are reduced to the study of uniform continuity in an appropriate metric by means of S-continuity in the nonstandard model (which was introduced by A. Robinson). The connection with the classical Picard theorem is made through a generalization of a result of A. Robinson on S-continuous :J!!.holomorphic functions. S-continuity offers considerable simplification over the standard sequential approach and permits a new characterization of these growth requirements. BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A BANACH SPACE is a nonstandard approach to the pre-dual Banach 00 space for H (D) which was introduced by Rubel and Shields. A new characterization of the pre-dual by means of the nonstandard hull of a space of contour integrals infinitesimally near the boundary of an arbitrary region is given. iv A new characterization of the strict topology is given in terms of the infinitesimal relation: "h g k provided II h-k !I is fini te and h( z) ~ k( z) for z E (*D) ". A new proof of the noncoincidence of the strict and Mackey topologies is given in the case of a smooth finitely connected region. The idea of the proof is that the infinitesimal relation: "h ~ k provided lfh-k!I is finite and h(z) ~ k(z) on nearly all of the boundary", gives rise to a compatible topology finer than the strict topology. v TABLE OF CONTENTS I. UNIFORM CONTTI>JUITY AND RATES OF GROWTH OF MEROMORPHIC F1JNCTIONS 1. Introduction • 1 2. A Mean Value Theorem in Metric Spaces . 2 3. Continuous ~:'-Meromorphic Functions 9 4. Invariant Normal Families 14 5. Milloux Circles and Points of Discontinuity • 20 6. AN ote on Two Cluster Set Theorems of Gauthier • 27 BIBLIOGRAPHY II. 25 BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A BANACH SPACE 1. In tr oduc ti on • • • 29 2. Introduction to H(D) and its Pre-dual P(D) 30 3. Embedding (P(D), H(D)) in the Nonstandard Hulls (Po(GLHo(G) ) . . . . . . • . . . • • . 32 4. The Weak-Star, Strict and Mackey Topologies 37 5. Non.coincidence of the Strict and Mackey Topologies . • • . . 42 BIBLIOGRAPHY APPENDIX 1: THE NONSTAL"'\JDARD HULL AND DUALITY 47 48 APPENDIX 2: SWEEPIT'JG MEASURES TO THE BOUNDARY OF SMOOTH REGIONS . • • • . 52 APPENDIX 3: NONSTANDARD MODELS 56 1 I. UNIFORM CONTil\!UITY AND RATES OF GROWTH OF MEROMORPHIC FUNCTIONS 1. Introduction: The purpose of this section is to show that several theories of the rate of growth of meromorphic functions can be treated in a unified fashion. Introducing appropriate metrics we see that they all amount to a study of uniform continuity (or uniform equicontinuity). the case of Julia-Milloux's theorem and Gavrilov's W p In classes (5. 6) the introduction of the metric is new. The connection between these theories and the classical Picard theorem is made through Theorem (3. 1) via the mechanism of Scontinuity. (This generalizes a result of Robinson [19 J who introduced S-continuity). In standard terms this relates exceptional values and non-uniformity, but the use of nonstandard analysis and S-continuity results in c _o nsiderable simplification because non-uniformity and sequences of Milloux circles are reflected in a single S-discontinuity at a remote point. In part, this a ls o extends Robinson's treatment of the holomorphic Julia-Milloux theorem to the meromorphic case. The nonstandard approach can greatly simplify a number of proofs of known results, particularily in the extens.i ve theory of normal meromorphic functions (4. 8). We offer a few simpl e examples. The original motivation for this work was the study of normal functions, unfortunately a search of the literature revealed that most of our early applications were done during the 1960's by various authors. 2 Some remarks about our setting are in order . Smooth convex metrics are emphasized because then we have a mean value theorem with which to measure S-continuity. simple result seems to be new.) (In this generality even this A more general discussion (without magnification operators) in terms of uniform continuity or S-continuity is possible. The domains of functions are viewed as Riemann surfaces for the sake of simplicity. Even in the case of a planar region we wish to emphasize a preferred metric and not be confused with others. This is important since different metrics have different infinitesimals and infinite galaxies. Moreover, some care is needed to apply Theorem (3. 1) at rem ote points, for example, in the plane metric near the boundary of the unit disk the infinitesimals are cut off by the unit circle, whereas Theorem (3. 1) applies with respect to the hyperbolic metric. We feel therefore that even in the case of studying boundary behavior of a function defined on the unit disk it is easier to think in terms of a Riemann surface. We hope that this will not cause con- £us ion; the primary examples are the punctured plane, the plane, the unit disk and hyperbolic surfaces (the punctured disk in specific). We shall use a leisurel y elementary style in___this section. In effect the point is that some of the known work is more elementary than it might appear because of complicated sequential arguments which our app roa ch avoids. 2. A Mean Value Theorem in Metric Spaces: A metric space (X, d) is said to be convex if for each A, B EX, 3 there exists C EX, different from A and B, for which d{A, B) = d(A, C) + d{C, B). When a metric space is complete and convex any tvvo points can be connected by a segment, that is, an isometric image of the interval [O, d(A, B)]. course.) (The segment need not be unique, of In what follows (X, d) will be a complete convex metric space. We give an adaptation to metric spaces of a proof of W. A. J. Luxemburg for a mean value theorem in Euclidean n-space. We begin with a lemma following a theorm of P. Levy. Let (Y, 6) be an arbitrary metric space. Let A, B EX and let [A, B] be a segment betvveen A and B. (2. 1) LEMMA: If f: (X, d) -> (Y, 6) is continuous, then for each natural number n ~ 3 there exist points A , B E {A, B) such that n n (1) d(A, A ) < d( B , A) ( 2) d(A , B ) n n n n = d(A , B) /n 6 (f(A), f(B)) 6 (f(An), f(Bn)) s {3) d(A, B ) PROOF: Let ~: [O, d(A, B)] _. [A, B] be the isomorphism from the interval in R onto [A, B] in X. co(t) Normalize to = ~(d(A, B) · t) , t E [O, 1 ] Define g ( t) = !!. ~( <p( t + ~) ) ' £( <p( t) )) 4 for fixed but arbitrary n ~ 3. Then by the triangle inequality n-1 .6( £(A), f ( B) ) 6 ~ a( k) k=O n o and 6( f (A) , f ( B) ) ~ 1 d(A, B) n n-1 6 g(k / n) k=O [ d(A~B)) Since the terms in the sum are non-negative, either (1) for every k, (2) for some k L(f (A), f (B) ) d(A, B) g(k/n) = d{A, B) /n or 0 .6( f (A), f ( B) ) , In case (1) pick k ---ci(A, B ) 0 < g( k /n) 0 d(A, B) /n f:. 0 or n-1 or in case (2) when k f:. 0 or 0 n-1 we take An= '10(k / n) and Bn = <P(k +1/n). 0 0 In case (2) when k 0 = 0 or n-1 we do the following. By con- tinuity of g(t) we may move t away from the appropriate endpoint while still maintaining the strict inequality in the expression (2). In 1 this case take An= <P(t) and Bn = cp(t+n). This proves the lemma. The magnification or metric derivative of a function f: X - Y can be described as follows. Let C be a standard point of >''··x whose monad is non-degenerate (i.e., a non-isolated point). Provided that for every pair of.points A and B within an infinitesimal of C, .6(f(A), f(C)) /d (A, C) is finite and infinitesimally close to 5 .6(f(B), f(C)) /d(B, C) we say the magnification of f at C is where st denotes the standard part homomorphism. In other words, as long as the standard part exists and is independent of the particular A within an infinitesimal of C, this expression is the magnification. We describe the operator M(.6/d) in the standard model by applying this definition at each standard point. As an operator M{.6/d) can be extended to the nonstandard model, though the above description does not apply for internal functions. We leave the limit definition to the reader since we shall not have any need of it. (2. 2) A Mean Value Theorem: Let (X, d) be a complete convex metric space, let (Y, .6) be a metric space, and let f: X ~ Y have a magnification everywhere on X. Then for every segment [A, BJ in X, there exists C E (A, B) such that ·6(f(A),f(B)) s; M(6/d)f(C) d(A, B) .. PROOF: f is continuous since d(x, y) ~ 0 implies .6(f(x), f(y)) ~ O. Thus we may apply the lemma as followso Pick A , B E (A, B) satisfying the conditions of the lemma, 3 3 next pick A , B E (A , B ) and proceed by induction. 4 4 3 3 Take the nonstandard extension of th_e sequences (An) n EN and (Bn) n EN. Let w be an infinite ~:~-natural number and let 6 C = std(A w) = std(B w), the standard point C such th~t Cg Aw g Bw. (A segment is an isometric image of a real compact interval.) Now we conclude by examining the following inequality. 6(f(A),f(B)) ~ 6 (f(Aw),f(Bw)) d(A, B) d(Aw, B w) d{A , C) [6(f(A ), f(C)) + o] w w ~ - - d..... (A.,...-,B~--)d~{A...,._-,--c-)- - w w ~ M(6/d)f(C) + where d ( C , B ) [ 6( f ( C), f ( B } ) + e J w w w d(Aw, B w)d(C, B w) s o, € and s are infinitesimal. We use continuity to see that 6(f(Aw}, f(C)) ~ L{f(Aw}, f(B w}), that is, that 6 and €are.infinitesimal. We use property {l) of the lemma and convexity to combine d(Aw, C) + d_(C, B w) = d{Aw, B w). This completes the proof of the theorem. REMARKS: A non-convex example where the theorem fails is provided by f(x) = x on the interval [O, 1] where the chordal metric of stereographic projection is taken in the domain space and the spherical arc length is taken in the range. TI I Then M(s/x)f{x) = 1 and s(f(O),f(l) >/x(O, 1) = I I 2 2 I ./2 = TI /2 4. > 1. As the following example shows, a minimal growth condition is not possible. f(x} = Ix I on [-1, 1 ] has magnification 1 even at zero whereas f{±l) = 1 so nowhere is the mag~ification below the divided difference~ Applications to meromorphic functions follow. 7 (2. 3) Some examples of complete convex metric spaces are: !Ix-y II. 1. A Banach space with the metric 2. The unit disk with the hyperbolic metric. 3. The Riemann sphere with the great circle arc length metric (or any mani fold with global geodesics). 4. The complex plane with zero removed and !log~ y I (principal value -TI< arg(z) ::;.; TT) as a metric. The following result is an application of Theorem (2. 2). See section 3 for the definition of S-continuity and Robinson [19 ]for more details. (2. 4) COROLLARY: If f: (X, d) - (Y, 6) is an internal map whose magnification exists and is finite at each point of the internal complete convex metric space (X, d), then f is uniformily S-continuous on X. PROOF: The set of bounds, B = [r E ~:~R +: IMf(x) I < r for every x EX} is an internal set since it is . described by an internal statement. Since B contains all infinite positive nonstandard reals it must contain a finite uniform bound for the magni fication of £. Now apply (2. 2). While we are on the subject of uniform continuity we give a result (which we use in (6. 1) below) that we hope gives the reader who is unfamiliar with nonstandard analysis an idea of the meaning of i nfinitesimals around a remo t e point. Sequences without limits in X, but which tend together, play a role analogous to infinitesimals around a remote point. 8 (2. 5) THEOREM: Let f: (X, d) - (Y, !::.) be a continuous standard function. The following are equivalent: (1) f is uniformly continuous on X. (2) >!'f is S-continuous on the rem ote points of >!'x, and hence 'X everywhere on ·· X. (3) Every pair of sequences (x : n EN), (y : n EN) which n n satisfy d{x , y ) - 0 also satisfy l::.(f(x ) , f(y ) ) - O. n n n n In particular when Y is compact, if f is not uniformly continu ous there are sequences (z n : n EN), (w n : n EN) such ~~ that d(z ,w) -o, f(z)-> a, and f(w) - b~a. -- n n n n PROOF: The equivalence of (1) and (2) follows from a --more general result of Robinson [ 1 9], but the fact that (1) implies (2) also follows automatically from interpreting the definition of uniform continuity in the nonstandard model and applying it at a remote point. That (2) implies (3) follows easily from the nonstandard interpretation of d( zn' y n) - 0, namely, tha t when w E #N, d{xw, y w) ~ 0. By S-continuity C:.(f(x ) , f(y ) ) ~ 0 and (3) follows. w w We can con.elude by showing that if f is n o t uniformly continuous (3) does not hold. points x n and y n There exists e > 0 such that for every n there are with d(x , y ) < 1 /n and .6(£(x ), f(y ) ) > E;, by the n n negation of uniform continuity. n n 9 3. Continuous ~:'-Meromorphic Functions: In this section we give a basic lemma on S-continuity for inter- nal or >!<:-meromorphic functions. (The reader is referred to Appendix 3 for the notion of a >!<-transform.) Since the theorem is local in nature and since we wish to consider different metrics even on familiar regions like the punctured plane and the unit disk it seems best to think in terms of coordinate patches on a Riemann surface. If Mer (0) denotes the standard space of meromorphic functions on a standard Riemann surface O, then >!<Mer is a function defined on >::_Riemann surfa ces. No harm seems likely if we extend >!<Mer to external subsets of a surface by requiring that this means there is a >!<-region containing the set and a function >!<-meromorphic in tha t region. Let O be a >!<-R iemann surface with a · t opolog i cally compatible >!<-metric d. We will say f E >!<Mer (0) is S-continuous in d provided it is S-continuous in the sense of Robinson [19] as a map from (0, d) to (:cs, s), the >!<-Riemann sphere with the great circle metric. At a E 0 this means that for every standard e > 0 ( e ER+) there exists a standard o > 0 such that d(z, a) < 6 implies s(f(z), f(a)) < e, or equival~ntly that z g a implies f(z) ~ f(a). Where the n o t ation denotes the respective infinitesimal relations. Now let (0, d) be a standard Riemann surface. Robinson [19] has shown that the near-standard >!<-meromorphic functions on >!<O (with respect to uniform convergence on compact subsets) are those which are S-continuous . on the near-standard points of >:<O , ns(:'o). (This can also be shown by writing down the monad of the uniformity of compact 10 convergence in >:'Mer( O) x >:'Mer(O), namely, (f, g) E µ(u~ if and only if f(z) ~ g(z) for all z E ns(:'rL). pact. We use the fact that 0 is locally com- See section II (4. 2) for more on monads of uniformities.) The standard part of a function can be taken pointwise (with respect to s) d df and dz ( s t(f) ) = st( dz). Now that we see the basic importance of S-continuity for ~:'-meromorphic functions, we give the following local theorem. We state the result in terms of the plane metric, p(x, y) = lx-yj, of a coordinate disk. For this reason some care is necessary in applications at remote points or when several metrics are involved. Let f be an internal meromorphic function defined on the monad of zero and hence in some finite disk of the complex plane. refers to the plane metric p as above. Continuity Magnification refers to M(s/p). (3. 1) THEOREM: The following are equivalent: (1) f is S-continuous at zero. (2) There exist three values a, S, y E ~:'s, finitely separated in the spherical metric, which f does not attain in the infinitesimals, o. (3) The magnification of f, Mf(z), is finite on the infinitesimals, o. PROOF: The equivalence of (1) and (2) follows from a theorem of 11 Robinson [19, Theorem 6. 3. 1] and the following lemma: ~·, The near standard transformations of the ·:.Mobius transformations are those specified by taking three finitely separated points on ~:~S onto three other finitely separated points on ~:~S. One way to see this is by first observing that the infinitesimal group is described by taking three finitely separated values to their respective standard parts on S. ~·, Direct computation: a., S, y E .~S and * s a., etc. a., y E C, then when a = [w,a;b,c] = [z,a.;p,y] implies w-a w-c x~ 1 . z-a = xz-y' where Then w = [(a - xc) z + ( xac - a y) ] / [ ( 1 - x} z + ( xa- y) ] so that w e = (1+6)z+ 'r)z+ 1 where 6, e, n are infinitesimal. A transformation of this last type changes finite values only by an infinitesimal and leaves infinite values infinite so it is within an infinitesimal of the identity. Conversely, an infinitesimal transformation moves each point at most ans-infinitesimal. 0, 1, oo The pre-images of will therefore uniquely determine the transformation9 One may now apply Robinson's theorem to the mapping g where =w O f w is the ~::Mobius transformation taking a, S, y of (2) into 0, 1, Next we show that (1) implies (3). Assume f(O} = 0, otherwise work with w of where w is a ~::_Mobius rotation of the sphere taking f(O) to zero which is justified since M(s/p)f = M(s/p)wo £. [E.g., w(z) = z+f(0)/1-f(O)z.] By continuity and the fact that f is defined on 00 • 12 a standard disk around a, we know If I is smaller than 1 on a disk of standard positive radius, lf'( b) I r, about a. Thus, integrating around = I0/2 rri) J (f{z)/ (z-b) 2 )dzj ::;; l / (r / 2) 2 , and M(s/p)f(b) is finite on o since Mf{b) for b ~ 0 , = Ir' (b) I /(1+!f{b)j2). For z ~ b, The magnification M(s / p} is usually denoted by p and is called the spherical derivative. This shows (1) implies (3). We conclude by remarking that since the segment from 0 to b in o is internal, the set [K: pf( z) < K , z on the segment} is internal and contains all infinite numbers, so pf(z) is finitely bounded. By Theorem 2. 2, s(f(b),f(a) ) ::;; K !b-a! ~ 0 , 0 so f is S-continuous and (3) implies (1). We wish to deal with applications of this theorem at remote points of a given metrice In these cases it is necessary th at .·~-. e >'' cl-infinitesimals 11 look 11 like the monad of zero in .~C. reformulation of '( 3.1) follows. The precise 13 (3. 2) We begin with an example of the difficulties one may encounter (which also shows why we view domains as Riemann surfaces). Take the right half-plane [z EC :Re(z) > O} with the plane metric !x- y l and consider the function exp(l /z) in the part of the monad of zero which lies in the right half plane. The function omits the entire unit disk and still fails to be S-continuous. We take O a ~:~-Riemann surface, ~:~ U is the •.-.~, unit disk, ri. z E >:~ C : (3. 3) d a >!~-metric on O and b E O. I z I < 1 }• DEFINITION: We say b is the center of an S-disk of O with respect to d provided there exists an inte rnal conformal (1-1) mapp i ng <p:>:~U - O satisfying: (1) co(O) = b, (2) cp( o) = od(b), {3) M{d/p)<p(z) exists, is finite and non-infinitesimal for z in the monad of zero. (3. 4) COROLLARY: The conditi o ns of Theorem (3. 1) apply at the center b E 0 of ~ S-disk with respect to 'd where S-continuity in (3. 1. 1) and the magnification in (3. 1. 3) are taken with respect to d. PROOF: f{<p(z)) is S-continuous at zero if and only if f(w) is d-Scontinuous at b and M(s/p)f ocp(z) = M(d/p)<p(z) · M(s/d)f(w). (Existence of the latter magnification is imposed by (3. 3. 3).) 14 4. Invariant Normal Families: The purpose of this section is to unify several theories of the rate of growth of meromorphic functions whose definitions involve normal families constructed from given functions. These theories are discussed and applications of the basic theorem are given at the end of the section. We feel that for a number of applications S-continuity applied directly is simpler than introducing normal families and this is justified by Corollary (4. 3). We shall discuss a basic setting which is less general than is possible, but which encompasses the three classical cases given in the applications. One generalization is mentioned in (4. 5). We feel that a number of results have been unnecessarily obscured in the classical theories and we feel that S-continuity can reveal their true simplicity. Our basic setting is as follows. 0 is a Riemann surface with a topologically compatible convex metric d. We assume that d is asymptotic to the plane metric of a coordinate disk D at its center a ED c 0. In terms of magnification operators this means M(d/p)id(a) = l, where id denotes the identity function and p the plane metric of n.· W is a group of conformal d-isometries of 0 onto itself such that for every b E O there exists a w E W with w(a) As a result of this, = b. dis a smooth metric, M(d/d)w(z) = 1, for every w E W and z E 0, and the magnification of a meromorphic function M(s/d)f(z) exists· at each point of O (since d is asymptotic to p at a). 15 Let F be a subfamily of Mer(O). We say F is W-invariant provided FoW = [fow: fE F and wEW} = F. When W consists of all conformal automorphisms of O, we say F is conformally invariant. If F is an arbitrary family, F OW is invariant by the group property of W. THEOREM: (4. 2) The following are equivalent for a W-invariant family F: (1) F is a normal family. ( 2) M ( s Id) f ( z) is a finite for eve r y z E ··~ 0 and eve r y f E (3) M(s /d)f(z) < K (a standard constant) for every z E 0 and *F. f E F. (4) Every f E ':'F is S-continuous on all of >:~O and hence uniformly S-continuous in the metric d. PROOF: (1==>2). Take f E '~' F and z E >~ 0. >'< Let w( a) = z, w E 'W. =M(s/p)g(a) = M(s/d)g(a) is finite for every g E >!'F by Robinson's pg(a) characterization of normal families [19, Theorem 6. 4. 1] and Theorem (3. 1). M(s/d)f(z) =[M(s/d)f(z)} [iv1(d/d)w(a)]=M(s/d) f ow (a), so (2) holds. (2=:>3). z E >!< >!< Since the set of bounds of Mf(z) where f E F and O is internal and contains all the infinite positive numbers there is a standard bound K. Condition (3) holds since its *-transform holds (with this K) in the nonstandard model. (3 =?4L Apply Corollary (2. 4). 16 (4~1). S-continuity on the near standard points implies that Fis a normal family by Robinson [ 1 9, Theorem 6.4.1 ]. (The topo- logical compatibility of d enters as st(x) = st(y) if and only if d(x,y)~O, (4. 3) ,,, forx,yEns('o).) COROLLARY: When F = [£ , ••• , fk} OW the conditions of the theorem are 1 equivalent to (standard) uniform continuity of the fk on all of O. In the setting of (4. 8) this is apparently due to Lappan [12]. (4. 4) REMARK: Condition (3) is a generalization of results of Yosida, Noshiro and Lehto-Virtanen. ( 4. 5) One generalization of this idea is to take W to be conformal mappings defined only on D and such that for every b E O there is a w E W with w(a) = b. M. F. Behrens has obtained some results in this case where O is a disk with holes removed. tion of a fixed disk translated within O. W consists of dila- A function f for which f o W is normal is termed regular and our condition (3) states that dist(z, b 0) pf(z) is bounded. Examples: ( 4. 6) Julia exceptional functions. Take O = C \. [O}, the punctured plane, W = [bz:bEO}, d(x,y) = llog~j where -rr<arg ~rranda= 1. y Ostrowski [18] discovered that Julia's theorem fails for meromorphic z-2n functions, e.g. , take f(z) = II - - , then f o W is normal. z+2n 17 Marty [1 5] character i zed Jul i a exceptional functions as those for whi ch Iz I pf Iz I is bounded. ( p denotes the spherical derivative, M(s/pL see the proof of (3.1).) This is condition (3) since M(s/d)f(z) = I M'(p I d)w(l) pf( z) = z r pf{ z) where w(l) = z. ::: : M ( P Id) W ( 1 ', :;:; This is because when x ~ 1' Iz I I x - 1 1 :-;::; :::-: I z I , ana t ne f ar s1. d es are s t an d ar d num b ers, I log x j J 1 hence equal. (4.7) Yosida's theory [20]. I Take O= C, the complex plane, W = [z + b: b E O}, d(x, y) = x-y I and a = O. (Doubly periodic functions arise in this context with 0 as the universal covering surface of the torus.) Yosida obtained condition (3) as necessary and sufficient for normality of f oW in the form pf(z) < K. (3.1) that M(s/d)f(z) (We showed in the proof of = pf(z) = jf'(z) I/ (1+jf(z)1 2 ).) Yosida also observed that results similar to Julia-Milloux's theorem hold in case normality fails. His results follow from our work in section 5 below. Yosida also connects this growth requirement with the Nevanlinna characteristic by integrating pf(z) in the Ahlfors-Shimi.zu formula. (4. 8) Normal meromorphic functions: (Noshiro [16 ], Lehto-Virtanen I z I < 1}, the unit disk, W = {exp(iB) [( z-a) /(az-1) J: e E R' Ia I < 1}' all conformal automorphisms [14 ].) Let O = U ~ [z E C : of U, a= 0 and d(x, y) = n (x, y), the hyperbolic metric (= (1 /2) log[(jx-yl + lyx-11)/( !x-y! - lyz-1 I)]). A meromorphic function is called normal if f OW is a normal family. 18 When G is a hyperbolic surface so U is its universal covering /\ surface, a n1e r om or phic f unction on G is normal provided f normal on U where P is the projection of U onto G. = f OP is This definition is extrinsic to G but conditions (2), (3) and (4) apply directly by projecting the metric. The S-continuity approach on G could be applied directly. Noshiro [16] gave condition (3) in the form (1-zz) pf(z) < K. know M(s/n)f(z) = M(s/n)fow(O) = [M(p/11)w(O)]pf(z) = (1-zz) pf(z), where w(O) = z. This is because M(p/ T) )w( 0) ""' T)(x,O)] [I ((x+z) /(Zx-1)) - z We !/ = [(x+z-z(zx-1))/(zx-l)T)(x,O)] ~ (l-zz)(!xl/11(x,O)) ~ (1-zz), where x ~ O. Since the far sides are standard and within an infinitesimal they are equal. (The hyperbolic metric is asymptotic to the plane metric at zero, so (!xi /n(x, O)) ~ 1.) Applications of a classical nature to the theory of normal functions are possible and the author has given a number of simplified proofs of known results using the nonstandard theory of the metric space (U, n). We give a few examples of this nature which involve a minimum of function theory. If aE ~:<U, then-galaxy around distance from a, G(a) a is the set of points a finite = {z E >:<U: n(z, a) E O}. We begin with an observation of N oshiro [16] which the reader can easily generalize to other settings (see Yosida [20]). (4. 9} If <,0(t), 0 ~ t < 1, is a continuous curve in U with I <,0(t) I -> 1, and if the normal func tion f satisfies f(cp(t)) -> b as t __, 1, then f OW has the constant function b as a limit. 19 Let a~ i and a,= <P(a) be fixed. Take w(z) = [(z+a) / (a,z-1)]. The function fow is S-continuous on ns(U) by (4. 2), hence st(f ow) is meromorphic. Since f(<p(t)) --> b, whenever t ~ 1 we have f(w(t)) g b, therefore st(f ow) = b on the points which map onto the points <P(t) which lie in G(a,). This means st(f ow) is constant on a set with a (non-trivial) adherent point, hence identically constant. f ow~ b on all of ns(:~U) = G(O) and f is near constant on G(a,). Also, In fact we see that a necessary and sufficient condition that f o W has a >'~ constant limit is that f is near constant on a galaxy of · U. (4.10) Nonstandardizing the work of Hoffman [10], M. F. Behrens [2 J has shown that by identifying infinitesimally near-by points of infinite galaxies which contain points of interpolating sequences the galaxies correspond to non-trivial Gleason parts of H the pseudo-hyperbolic metric to the parts metric. 00 (U). Moreover, I(x-y) /(yx-1) I is infinitesimally close Now since a normal function is S-c o ntinuous on the galaxies, identifying points infinitesimally near-by and taking the standard part of the function gives us a standard function, continuous in the parts metric, defined on the G l e a son part. This proves a recent result of Brown-Gauthier [3] that normal {unctions can be extended to non-trivial parts. Many other applications could be given, but we refer the reader to the forthcoming monograph of A. J. Lohwater for m o re on normal functions. The bibliographies of N oshiro [l 7] and Collingwood- Lohwater [4] contain many other interesting references to normal functions. 20 5. Milloux Circles and Points of Disc on tinuity: Milloux's theory of vcercles de remplissage' has seen recent interest with generalization in several directions (Lehto [13 ], Lange [11 ], Gavrilov [8], [9] and Gauthier [6], [7]). Robinson [19] "nonstandardized 11 the classical theory obtaining a new lemma for the existence of such circles. Robinson's lemma does not generalize directly the meromorphic functions because the >:~-sphere has only one S-component. The main idea of replacing sequences of circles with the monad of a discontinuity does extend to a very general setting as we show in this section. Several of the known results in the various settings reduce to the equivalence of (3. 1. 2) and {3. 1. 3) at a discontinuity for an appropriately chosen metric. Our method simplifies the previous approaches t o the theory and we hope also shows how nonstandard analysis can be useful when complicated quantification arises. This approach a ls o shows that ' 1Julia-sets 11 and 11 Milloux-sets 11 amount to the same thing since they both reduce to a discontinuity. In a metric space (X,d) we shall use the notation and where B c X. d(x, b } < 8} ' . >:' -X. We also use the >.~-transform in od( B) -·~ = [x E .. X :3 b E B and Also d ( x, b) R:J 0 } shall denote the infinitesimal neighborhood of B. 21 (5. 1) DEFil\HTIONS: We say A c O is a d-Julia-se t for f if for every (standard) positive e, f(D d(A; e) ) 2 S \[a, B} , for two values ex, BES, the sphere. (J-set.) o, We say A is a d-Milloux-set for f if for every positiver, €, there exist s1 , s2 ES and y EA such that d{a, y) > r and (M-set.) The connection between discontinuities and standard J- and Msets is as follows. (5. 2) THEOREM: Let b be the center of an S-disk on O with respect to d. If b is an S-discontinuity in the metric d for a standard meromorphic ''< function f defined on O and if b E od(' A), then A is a J-set. If b is in an infinite galaxy, then A is an M-set. PROOF: We may apply (3. 4) at the discontinuity to see that for at most two standard ex, 13 E S. ...,, ~'-- We know that the set of standard points in ~. f( .~ [D d(A; e) ]) is f{Dd(A;e)). When e is standard od{b) E.. >!~Dd(A;e) = Dd(A;e) and 22 therefore A is a J-set. If b is infinite the standard set B given below satisfies ~:<B .2 o+ X (<:R \ o) x (:<:R \ o), that is the first component can be any finite positive real number and the next two any non-infinitesimal. Then 0 ,1, ('"B) = B = R + X R + X R' and A is an M-set. ~ s • t. d ( y, a) > r (If r is finite and 6 and e non-infinitesimal we take yE o,(b) Q ''<: n ''A and apply the reasoning in the first part of the proof.) (5. 3) REMARKS: When we begin with a given remote S"'.'discontinuity of a standard function we may obtain a standard sequence which is a J-set or M-set by applying (2. 5). (This avoids the somewhat more delicate problem of approximation of a particular point by a standard sequence.) When w E #N, xw of the sequence in (2. 5. 3) is a (remote) S-discontinuity. . ''<: If d is finite exactly on ns( O), as is the case in (4. 6), (4. 7) and (4. 8) for example, then J-sets and M-sets coincide for standard ~·- functions since they are necessarily continuous on ns('~o). J-sets and M-sets have infinite discontinuities in their nonstandard extensions, so their non-existence in the examples (4. 6), (4. 7) and (4. 8) is equi valent to normality. Next we extend a result of M a. rty-Lehto to this setting. Ac 0, let If 23 lim sup h( z) z EA = inf [sup(h(z) : z E A and d( z, a} > r) : r > 0] , for real valued functions h(z), where a E 0 is fixed. We shall also assume from now on that if d(z, a) > r for some fixed finite r (5. 4) 0 0 (z E >~O) , then z is the center of an S-disk. THEOREM: If lim sup M(s / d)f(z) = oo, then >:'A contains an infinite SzEA discontinuity of f, or A is an M-set. PROOF: The >:~-transform of lim sup says there is an infinite point .... . z E ···A for which M(s/d)f( z) is infinite since we have assumed od(z) i s an infinitesimal disk we may apply (3. 4) which says then that z is a point of discontinuity. As applications we now consider some of the known results. (5. 5) Julia-Milloux theorem for meromorphic functions: Apply (5. 2) . d to the setting of example (4. 6). Observe that since x y in that = !x-y j < 6 !y I for some infinitesimal 6, w_e may substitute the J- and M-set conditions for standard disks Ix-y I < oIy I metric if and only if which is the class:lcal form.. if lim sup Thus we have the classical result that Iz n I pf(z n ) = oo then the sequence (z n) -is-an- M-set for £. We have already remarked that ! z j pf(z) can fail to be infinite in ( 4. 6). 24 (5. 6) Gavri l ov's classes W around oo. p (p :2: 1) ( [8 ], [9 ]) : 0 is a punctured disk The metric is given locally by the differential form M(s /d)f{z) = Iz j 2-p pf(z). W p is the class of uniformly continuous meromorphic functions, which was introduced by the requirement that lim sup Iz 1 2 -P pf(z) < oo. By examining 2 infinitesimals, d-disks may be replaced by I x-y I < E: I y 1 -P. The theorem of Gavrilov which follows is immediate. If I zn 1 2 -P pf(zn) a ne ig hborhood of point z n = oo, for a holomorphic function f defined in then for every r > 0 and e > 0 there exists a oo, such that in the disk ! z-z value in the circle lw n I < s Iz n j 2-P, f takes on every I < r with the exception of a set of diameter less than 2/r. ( 5. 7) Lehto and Virtanen [14 J proved that no meromorphic function can be normal (in the sense of example ( 4. 8)) in the neighborhood of an isolated essential singularity. We put the hyperb.olic metric on a disk punctured at . oo and find M( s I il )£( z) infinite near oo. = Iz I log I z I pf( z) must be Hence there are the corresponding J- and M-sets in the hyperbolic metric around any sequence on which lim sup M(s/n)f(z) = 00 • We contrast this with the case of (5. 5) where functions can fail to have these sets (see (4. 6) above). (5. 8) Functions in the unit disk: The study of M-sets (in the context of (4.8)) for the unit disk originated in Lange [11 J and has had many contributors. w ·e give one very simple example (which follows from more refined work of Bagemihl and Seidel [l ]; also see Collingwood- 25 Loh.water's [4] bibliography) to indicate the 'flavor' of the theory. Let r. r and A be boundary paths in U and suppose f -+ a along :i'~ ....... If M(s/n)f is finite on a finite neighborhood of 'r, .,,D n (r; e), ( e ER+), then f is near a on any sub-neighborhood of an infinite point of ~:~r. From this we see that if A lies in such a neighborhood of ,1, either f -+a along A or .,,Dn (r; e) is an M-set. Hence, if are finitely separated boundary curves and f -+a .£:2. r, r, r and A then either f -+a .£:2. A or every mutual neighborhood of the curves is an M-set. In particular, if f is normal, f -+a along I\ . 6.. A Note on Two Cluster Set Theorems 0£ Gauthier: In Gauthier [5 J a standard version of the following definition is given. U is the unit disk with n the hyperbolic metric. ,,, of n-infinite points of ···u, (or the set [z: sets A and A 1 2 Iz l ~ 1} ). _:::. U are equivalent if on (I I is the set Two standard ~!< n A 1) We have immediately: (6. 1) THEOREM: . Let f : U -+ S be a continuous f unction. Then f is unif or ml y continuous on U if and only if for every pair of equivalent subsets of U, A 1 ~ A 2 , the cluster sets C(f;A ) and C(f;A ) are equal. 1 2 The theorem is strictly standard, the proof is nonstandard. PROOF: ( ~): The nonstandard characterization of the cluster set is: 26 ~ & c (f ; A 1 ) = s t s ( f ( I n '•'A 1 ) ) = s t s (f ( 0 n ( I n •' A 1 ) ) = s t s (f ( 0 = C(f;A 2 ). The step stf(I n ~:~A.) = st(f( o(I n ~:~A.) (j = J & (I n "'A 2) ) 11 1, 2) requires J uniform continuity. {<=:=): If f is not uniformly continuous there exist sequences (xn) n EN and (yn) n EN with n(xn' yn) - 0 and ri(O, xn) while f ( x ) - a f. S .-. f ( y ) • n n In this case [ x n oo, ri(O, yn) : n E N } "' [ y } , but n C(f;[x }) -:/:. C(f; [y }). n n (6. 2) A corollary which contains Gauthier's Theorem 2 is: Ameromorphic function f is normal on U if and only if C(f;A ) = C(f~A ) for 2 1 every pair of equivalent subsets of U. In order to establish his results Gauthier introduced the fol- lowing cluster set in standard terms: A Of course, if A 1 ,...., A .._t,.. we have C(f;A ) 2 1 A = C(f;A 2 ). Now if f: u- S is A S-continuous on "'A we have C(f;A) = C(f;A) and iD: light of (5. 1) above, ~ when f is meromorphic, either A is an M-set or C(f;A) = C(f;A). sharpens his Corollary 1 and Theorem 1. This 27 BIBLIOGRAPHY [l] F. Bagemihl and W. Seidel, Behavior of Meromorphic Functions on Boundary Paths, with Applications to Normal Functions, Arch. Math. 11 (1960), 263-269. [2] M. F. Behrens, untitled preprint, to appear. [3] L. Brown and P. Gauthier, or e:;. communication, to appear. [4] E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge Tracts in Math. and Phys. 56, Cambridge at The Univ. Press, London 1966. [5] P. Gauthier, The non-Plessner points for the Schwarz triangle functions, Ann. Acad. Sci. Fenn. AI 422 (19 68), 1 - 6. [6] P. Gauthier, 'Cercles de remplissage' and asymptotic behavior, Can. J. Math. [7] 21 (1969), 447-455. P. Gauthier, 'Cercles de remplissage' and asymptotic behavior along circuitous paths, preprint, to appear in Can. J. Math. [8] V. I. Gavrilov, The behavior of a meromorphic function in the neighborhood of an essentially singular point, AMS Transl. (2) 71 (1968), 181-201. [9] V. I. Gavrilov, Some classes of meromorphic functions characterized by their spherical derivative, I zv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 687-693. [l 0] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. Math. 86 {19 62), 74-111. 28 [11 J L. H. Lang, Surles cercles de remplissage non-euclidiens, Ann. Sci. Ecole. Norm. Sup. (3) 77 (1 960), 257-280. D. 2 J P. Lappan, Some results o n harmonic normal functions, Math. Z e it. [13 J 9 0 ( 1 9 6 5) , 1 5 5 - 1 5 9 . O. Lehto, The spherical derivative of meromorphic functions in the neighborhood of an isolated singularity, Comm. Math. Helv. [14] 33 (1959), 196-205. O. Lehto and K. I. Virtanen, Boundary behavior and normal meromorphic functions, Acta Math. [15] 97 (1957), 47-65. F. Marty, Recherches sur la reparti tion des valeurs d'une fonction meromorphe, Ann. Fae. Sci. Univ. Toulouse III 23 (1931), 183-261. [16] K. N oshiro, Contributions to the theory of meromorphic functions in the unit circle , J. Fae. Sci. Hokkaido Univ. 7 (1938)' 149-159 0 [17] K. Noshiro, Cluster Sets, Ergebnisse d. Math. new series 28, Springer-Verlag, Berlin, 1 960. [18] A. Ostrowski, Uber F olg en anal ytischer Funktionen und einige Verscharfungen des Picardschen Satzes, Math. Zeit. 24 (1926), 215-258. [19] A. Robinson, Non-S t andard Analysis, Studies in Logic, North Holland, Amsterdam, 1966. [20] K. Yosida, On a class of meromorphic functions, Proc. Phys. Math. Soc. Japan 3 ser 16 (1 934), 227-235. 29 IL BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A BANACH SPACE 1. Introduction: In this section we study a pre-dual for the Banach space of bounded analytic functions on a region which was introduced by Rubel and Shields [7 ]. section 2. Further introductory details are contained in We also recommend the expository paper of Rubel [5 J. In section 3 we give a characterization of the two spaces in terms of the nonstandard hulls of internal spaces of an approximating region. This allows us to approximate arbitrary regions by inter- nally finitely connected ones for the study of the space of analytic functions and leads to a representation of the pre-dual as a quotient space of the internal space L 1 of the boundary. In section 4 we introduce the topologies of the dual pair and rephrase a number of results in terms of infinitesimal relationships in the nonstandard extension. We also give a new characterization of the strict topology. In section 5 we give a new proof of a theorem of Rubel and Ryff [6] that the strict and Mackey topologies are noncoincident. We feel that this approach together with the approximations of section 3 may lead to a solution of Rubel's conjecture [5 J that these topologies are always noncoincident when a region supports nonconstant bounded analytic functions. This will be taken up in another place. 30 2. Introduction to H(D) and its Pre-dual P(D): In what follows D will denote an open connected subset of the Riemann sphere S (i.e. , D is a doma i n or region .) We will assume that D supports non-constant bounded analytic functions. H co The space (D) of all bounded holomorphic functions on D with the norm 11h 11 = sup( Ih(z) I : z ED) is thus an infinite dimensional Banach space over the complex numbers, C. (For short we denote H 00 (D) by H(D).) We wish to study the pre-dual of H(D) introduced by Rubel and Shields [7] from the point of view of nonstandard analysis. This leads to a new characterization of the pre-dual when D is infinitely connected though our opening remarks apply to the general case. The basic idea is to replace D by a region bounded by a >:<-finite number of smooth curves infinitesimally close to the boundary of D. We represent the pre-dual in terms of boundary integrals on the approximating region. A brief explanation of the >:~-transform of a concept is given in Appendix 3. We will use this idea freely to extend our vocabulary to a nonstandard model. 1 A smooth curve is a non-degenerate C -curve, meaning that if y is a parametrization, yr is a non-zero continuous function from an interval of positive length. A >:<-smooth curve is the corresponding internal notion. ( 2. 1) We shall say a >:<-region G approximates D provided that G is >!<-finitely connected, has a >:<-smooth boundary, and satisfies ns(<D) ~ G .:::_ ~:<D.· (nsC:<D) is the set of near-standard points of >:CD). It is always possible to approximate a standard region in this sense, 31 since, f or exa mpl e, we may t a k e an i nf i ni t e l y subscripted member of the n onstandard e x ten si on of an exh aus t ion of D. For these prelim- inaries, further assump t i ons w ill not be made about G, though we expect judicious choices to yie l d more information about the standard spaces. We denote the internal space of all internal bounded holomorphic ...,,,. ... ... functions on G by H(G) and similarily '''(H(D) ) by H("'D). This is con- sistent with the usual notation if we view H as a function defined on regions. Since >:~D ~ G, functions in H(~D) are mapped into H (G) by /".... restriction and the standard functions (elements of H(D)) suffer at most an infinitesimal reduction in norm. Complex Borel measures with support in D, denoted M(D) , are in duality with H(D) by the pairing (µ, h) = J h(z)dµ.(z) • D The measures which annihilate H(D) are denoted by N(D) = [µ E M(D) : ( µ., h) = 0 for all h E H(D) }. M(D) with total variation Iµ I(D) as norm is a Banach space and N (D) is a closed linear subspacee Rubel and Shields [ 7] have shown that the continuous dual of M(D) /N(D) is H(D). Their proof cons i sts of showing that an arbitrary measure has a represent ative which is absolutely continuous with respect to two-dimensional Lebesgue measure and of decreased total var i ation. 1 1 As a consequence L (D) /N (D) is also isomorphic to 1 1 M(D)/N(D) as a pre-dual for H (D ), where L (D) is the Lebesgue L 1 space and N (D) is the set of null functions for H (D), 32 [f E L 1 (D): J f(z)h(z)dm(z) = 0 for all h E H(D)}. This also show r;; D that t he pre-dual is separable. Rubel and Shields [7] also show, in the case of the unit disk, that measures can be swept to the boundary. Their argument is extended to finite l y connected regions, G, with smooth boundary, bG, in Appendix 2. In this case we have the Banach-isomorphisms We shall refer to the pre-dual as P(G) in any of these three roles when G is finitely connected and has smooth boundary, and P(D) for arbitrary regions in the latter two roles. As a function defined . on regions P extends to ~:~-regions in the nonstandard model via the ~:~-transform of its standard characterizations. As a result when G approximates ~:~D we have the above three internal characterizations of P(G) as either ~:~-Borel measures in G, or ~:'-L 1 or ~:'-L -functions on bG. 1 -functions in G, The last of these has no standard analog in D, generally speaking, and allows us to apply the Cauchy formula, etc. to these situations. We connect the internal spaces P(G) and H(G) with the standard spaces P(D) and H(D) by means of a nonstandard hull of the internal_ spaces. The general construction and the fact that the construction is consistent with the duality is contained in Appendix 1. The next section contains some basic results in the specific case. 3. Embedding (P(D), H(D)) in the Nonstandard Hulls (P (G), H (G}): 0 0 We form the nonstandard hulls of P(G) and H(G), which we 33 den ote by P (G) and H {G), respectively. 0 0 These are standard Banach spaces and the dual of P (G) conta ins H (G) by the general results of 0 0 Appendix 1. (3. 1) THEOREM: The standard space H(D) is properly norm embedded in H (G). 0 PROOF: The standard space H(D) is norm embedded in H (G), since a 0 standard function, extended via >:~ to >:~D, is only infinitesima~ly reduced in norm by restriction to G. An example of a function in H (G) which does not correspond 0 to any standard function is zA, where A is an infinite natural number. ,,~ ,,, and G = '• D = '•'-unit disk. The function z A ha.s norm one, but point- wise standard part zero. Since we have chosen G with an internal non-degenerate smooth boundary, every H (G) will contain such functions (even if H(D) 0 = C). To see this we only need to consider the Riemann mapping of the inside of one boundary component of G onto >:~U. Specifically, let a E ns(:~D) and p: G -+ >:~U be a conformal mapping of the inside of one boundary f'.J ,,,.. >'c component of G, G .2 ns("'D), onto 'U such that p(a) = 0. The function g(z) = p\z) has all its derivatives of order less than A equal to zero at a. Since g is S-continuous, being bounded, and since derivative commutes with standard part, g ~ 0 on ns(:~D). we als 0 know that Ig ( s) I = 1 on bG' s0 11 g II = 1. (Cf. Robinson [4].) This proves ( 3. 1). Our next result shows that remote elements of finite norm, 34 such as those above, play a role in determining the standard elements of the pre-dual P (G). 0 (A function which is non-near-standard in a cer t ain topology is termed remote.) (3. 2) THEOREM: The standard space P(D) is norm-embedded in the nonstandard hull P (G) of the approximating region G as the classes of those 0 norm-finite elements p E P(G), 11p11 E O, which satisfy: all norm-finite h E H(G), II~ (p, h) ~ 0 for 11 E 0, such that st(h) = O. PROOF: We first show that the natural embedding of P(D) into P (G) 0 induced by restriction of standard measures has the desired properties. /'.. µ E M(D)' (µI G) (B) = µ(B n G) denotes the restriction which is in M(G). Since D is cr-compact, Iµ I is a regular Borel measure and consequently Iµ I GI (G) = Iµ! (G) ~ Iµ I(D), because Let G .2 nsC:'D) and ns(:'D) is the union monad of the compact subsets of D. Now let e ER+ be a positive standard number. K, a standard compact subset of D, for which Now II I ,; 11 II I hdµ G h d I µI + G\:'K ~ .Z + Iµ! (D) sup( !h(z) I: z E ~:'K) There exists 35 when 11h11 is finite. If st (h) == 0, the second term is infinitesimal, I(µ I G, h) I < e for any standard e ER+. making The fact that the norm is reduced at most an infinitesimal can /'. be seen as follows using the fact that M(D) is norm-embedded in As a measure in MC:'D), I since Iµ, - (µ,I G) (:'D) ~ 0. µI G is norm-infinitesimally-near µ,, If V is equivalent to µ,I G for H(G) as an element of M(G), by extending it to be zero off G we have the equivalence of \) and µ,I G in MC:'D) for HC:'D). Taking nonstandard hulls this simply means 11 [µ,I G] II in P (G) equals 11 [µ,] 11 in P(D), because 0 11 µ, + A II ~ 11 µ l G + A II £or all A E N c:' D). Now we focus our attention on the converse: if p E P (G) w ith finite norm and satisfying the property of the theorem, then p corresponds to a standard pre-dual element. Fix e ER_+, a standard positive real number. Consider the internal family J of subsets F £ M('D) defined by the internal statement: 11 F E J< provided F is a >:'-finite subset of MC:'D) and if for each f E F we have I(£, h) I !5: 1 and 11 h II !5: 1, then I (p, h) I < e. 11 Every /'.... ~:'-finite set F for which °F ~ M(D), or F .2. M(D), is in d by the hypothesis of the theorem. A each r ER and whence /'.... To see this observe that r o E M(D) for x A ox equal to unit point mass at a standard point x ED, I (£, h) I ~ 1 for all f E F onl y if h is infinitesimal on D. Since 11h11 E 0 implies h is S-continuous on ns('D) we have that h is infinitesimal o·n ns(:'D) or st(h) = 0. A result of Luxemburg [2, Theorem 2. 7. 11 J states that in 36 ~ sufficiently saturated models (see Appendix 3) standard set whenever ·""- which contain M(D). ~ must contain a is internal and contains all ~:~-finite sets This means there are finitely many pre-dual ./"... elements p , •.• , pn E P(D) such that whenever 1 i = 1,2, .•. ,n and !lhl/ ~ 1 then I (p,h)! < e:. I (pi' h) I ~ 1 for For convenience we assume the p. are linearly independento l The remainder of the argument follows Luxemburg [2, p. 84, part (c) => (a)]. On the internal space K £ H(G) given by K = [k: (p., k) = 0; j = 1, .•. , n} J we have that the norm of the functional (p, · ) is less than e:. ~:~-transform By the of the Hahn-Banach-Bohnenblu.s.t -Sobczyk theorem we may extend this to a functional (in H(G)') (cp, · ) on all of H(G) with norm less than e:. For k EK, (p-cp , k) = 0. A simple induction argument shows that n p-cp = ~ c.p. • i=l l l Thus, II P -~ c . P . II = 11 <P II < e: • l 1 Another induction argument shows that c. are finite since 11p11 and l !!Pill arefinite(i=l, ... ,n). Let q =~ st(c.)p., then 11p-q11 < e: l l and p is norm-n·ear a standard element of P(D) since we may approximate it to within a standard e: by a standard q. 37 (3. 3) COROLLARY: 1 f E L (bG) with finite norm corresponds to a standard pre-dual e.lement provided that whenever ! 11 h I E 0 and st(h) = 0, then J f(z)h(z) !dzj ~ 0. bG There is a connection between this theorem and the completeness theorem of Grothendieck. Also, in case G is standard, the result specializes to a consequence of Luxemburg [2, Theorem 3. 1 7. 2] by the characterization of the weak star infinitesimals given in the next section. 4. The Weak-Star, Strict and Mackey Topol og ies: In this section we introduce the topologies of interest to us in addition to the norm topology for H(D). involves the weak topology on P(D). The Mackey topology also Much of the work of Rubel and Shields [ 7] and Rubel and R yff [ 6] involves the study of these topologies and their relation to a number of classical problems in function theory. A survey of the standard theory can be found in Rubel [5 ]. We shall give a nonstandard account of various known results mixed with a few new results. Our point of view is that of a uniform space in the sense of Bourbaki [l ]. basic nonstandard treatment. Luxemburg [2] has given the If (X, u) is a uniform space with uni- formity u, then the intersection monad of u, is an external equivalence relation. µ(u) = nu= n(:~U: U Eu), Conversely, if µ 0 is a monad in >:~X X ~:~X and an equivalence relation, it determines a uniformity _ for X. We shall write x g. y for x is within au-infinitesimal of y 38 which means (x, y) E µ(u). Also, the set of u-infinitesimals around x, µ(u)[x] = [yE ~:~X:{x,y) E µ(u)} = ou(x) . We caution the reader that o (x) is not necessar i ly a monad, which by definition must be the interu A section of the standard sets in a standard family of sets, n J. If X is a vector space, u is a compatible uniformity if and only A if o (x) + o (y) c o (x+y) for every x, y E X and o( A) o {x) c o (Ax) for u u-u u-u A A every A EC and x EX. We have in fact: o (x) + o (y) u u AO (x) = o (Ax) for every x, y E ~:~X and finite A. u u = ou (x+y) and The uniformity is Hausdorff if no two standard elements are within an infinitesimal and locally convex if o (x) is (externally) convex. u If the pseudo-metrics ( 'Y: 'YE r) characterize u in X, U x = y if and only if 'Y(x, y) ~ 0 for every 'YE r. A then The following charac- terization of weak-star infinitesimals follows from this statement. (4. 1) THEOREM: In H(:~D), h is within a weak-star infinitesimal of k, h g k, if /'. and only if (µ, h) ~ (µ , k) for every standard µ E M(D). /'.. Since the point masses are in M(D) for standard points of D, if h ~ k, then h( z) ~ k( z) for z E D. if in addition 11h-k11 is finite, then h(z) ~ k(z) for z E nsC:~D) by Robinson's result that finitely bounded functio n s are S-continuous. (4. 2) THEOREM: There are S-discontinuous a-infinitesimals. 39 PROOF: We construct an S-discontinuous a-infinitesimal as follows. Let P be a >!<-finite set which con t ains all the standard pre-dual ele- 1 -·~ /".. ments P(D} .:::_P q =/. st(q). If 6 q 1 c P(''D). Let q be a near standard point for which denotes the point mass at q we see that /'. [o ] ¢ P(D), since there are standard functions which are one-to-one q at st( q). p 1 EP 1 Now a function h E H(:'D) such that (p , h) 1 = 0 for every and ( [oq], h) = 1 is a a-infinitesimal, h g 0, and S- discontinuous at st(q). Let J be a family of complex valued functions defined on a Let 6 be a collection of subsets of X. set X. (4. 3) THEOREM: The uniformity of uniform convergence on the sets of 6 characterized by its infinitesimal relation as follows: is f ~ q if and A only if f(s} ~ g(s} for all s E v(6) = u6, the union monad of 6. The easy proof is left to the reader. (See Bourbaki [l] for the standard versiong) (4. 4) COROLLARY: In the space H(:'D), the infinitesimal relation for uniform con- ver ge nce on compact subsets is: h ~ k if and only if h(z) ~ k(z) for every z E ns(:'D). PROOF: The union monad of the compact subsets is ns(>!'D) by local compactness of D. 40 Now we can see that finitely bounded sets. a agrees with compact convergence on This was first observed by Rubel and Shields [7 J. (4. 5) SUMMARY OF RESULTS ON THE STRICT TOPOLOGY: The strict topology which is the topology induced on H(D) by the notion of bounded sequential convergence (h n -> h if I !h n 11 ~ M and hn -+ h pointwise) is also the finest t opology which agrees with bounded sets. are equivalent: We denote the uniformity by ~ a ~ and then the following (Rubel, R yff, and Schields [6 ], [7 J) a. h ~ k. b. (p, h) ~ (p, k) for every p E cmpN(P(~D)), the norm-compact points of PC:~D) = u c~:<K: K is a standard norm-compact subset of P(D) ]. c. (p , h) ~ (p , k) for every standard norm-null sequence. n n d. f(z)h(z) ~ f(z)k(z) for every z E 'D and every f E C (-D) = 0 >'< /'... standard continuous functions vanishing off compact subsets of D. In particular, if 11 h- k II is finite and h( z) ~ k( z) for z E ns(<D) then d holds. A partial converse is possible, namely d implies that h(z) ~ k(z) for z E ns(:<D) since we may take a standard c -function 0 which is one at a given near standard point. (4. 6) This proves: THEOREM: All p-infi_n itesimals are S-continuous on ns(:<D). 41 We contrast this t o the weak-star infinitesimals, Theorem (4. 2). The external equivalence relation 11 h g k provided !I h- k 11 is f ini te and h(z) ~ k(z) for z E ns(~D) 11 is compatible with the linear structure in the sense that if h g h' and kg k', then h+h' g k+k' and for finite /..., /...h g Ah'. Moreover, the equivalence classes are convex. This is not a monadic equivalence relation however and there are norm-infinite (3-infinitesimals. The relation g is particularly natural and in light of the model-theoretic significance of discrete monads (as the best approximation by standard sets) we feel our next result is not without interest. (4. 7) THEOREM: The discrete monad of the set of g -equivalent pairs of elements of H(~D) equals the set of S-infinitesimals; µ( S) = µD( [(h, k): kg k}) = nC:~E: E is standard and ~:~E ~ g ] . In other words, g determines the strict topology. PROOF: The above remarks state that b =£ µ ( (3), so µD( b=) £ µ( B) and we need only show that µD( g_ ) .2 µ((3). Suppose that ~:~ E ::::i b "" = and that n EN, a standard natural number. In the nonstandard model the sentence 11 there exists f E c (~D) such 0 that 11h-k11 < n and If(zi(h( z)- k(z~ I < 1 implies (h, k) E ~'E," holds since we may take f(z) infinitely large on K a >:~-compact set containing 42 The same sentence without~:< on D and E must hold in the ns('D). standard model and this means 6 agrees with the filter of µD(g) on bounded sets. a Since S is the finest topology which agrees with on bounded sets, the proof is complete. Perhaps it is worthwhile to examine the monad of zero. b b have that µD( [h: h = 0 }) = µD( =) [O] = o(3(0). compatibility of ( 4. 8) We This follows from the g with the linear structur.e. The final topology for this section is the Mackey topology on H(D) which is the finest whose dual is P(D). m(H(D), P(D)) or just m. We denote it by The Mackey-Arnes theorem states h m =k if and only if (p, h) ~ (p, k) for every p E cmp (PC:~D)), the weakly w compact points of PC:~D). m = (3 if and only if cmpN(P(D)) = cmp (P(:~D)), that is, if w and only if P(D) has the "Schur property" that weak and norm compactness coincide~ Rubel [5] conjectures that m =f. (3 so long as H(D) f; C. We always have m finer than 6, so one only needs to show s triC t incl us ion. 5. Noncoincidence of the Strict and Mackey Topologies: In this section we give a new approach to a theorem of Rubel and Ryff [6] that states that p(H(G),P(G))-:/. m(H(G),P(G)) when G is a (standard) finitely connected region with smooth boundary, bGe 43 We do this by introducing the infinitesimal equlvalence relation that two functions are finitely bounded in difference and infinitesimally close on most of the boundary. This relation give$ rise to a standard topology strictly finer than \3 and coarser than m. We do not know if this new topology equals the Mackey topology and the generalization of these results to the nonstandard hull of H(G) where G approximates ~:~D remains to be done. (5. 1) DEFINITIONS: a denotes the uniformity associated with the L 1 (bG)-norm of the non-tangential boundary values of functions h E H(G), Ih Ia = s = sup( a., p)' the uniformity generated by the entourages u n v' where U E a and V E \3. r is the uniformity associated with the finest topology which agrees with the topology of s on bounded sets. ~is the ex.ternal equivalence relation on H(~G) given by [{h, k) : II h-k 11 E 0 and h(z) ~ k(z) on most of bG}, precisely h ~ k provided 11 h-kl I E 0 and there exists T an internally measurable subset of ~:~bG for which h(t) ~ k(t), t E T and such that f Idz I ~ f I dz 1. T bG s and r actually came from an attempt to understand~ which is a natural refinement of g (and in The motivation to study a., fact a strict refinement as is easily seen). An understanding of ~ 44 seemed desirable from the beginning since ~-continuous linear functionals satisfy a dominated convergence theorem by a direct nonstandard argument. This hints at an integral representation and compatibility with the dual pair. We were unable to give direct extension procedures for such functionals to integrals, however. We have the following result. (5. 2) THEOREM: If h II h- k II is finite, then h g, k if and only if h ~ k. Moreover, g k and h g, k if and only if h ~ k. PROOF: If h ~ k then s Ih( z) - k( z) I Idz I ~ J Ih( z) - k ( z) ! Idz I + ' II h- k II J ~:'bG Idz I ~ 0 . ~~bG \ T T Observe that there is an infinitesimal e such that for all t E T, Ih(t)-k(t.) I < e since T is internal. This . proves h a= k. Next, if !Ih- k II E 0 and if J Ih( z) ~ k( z) I Idz I = 6 ~ 0, then jh(t)-k(t) I > r ~ 0 on T with finite positive measure leads to a contradiction, hence h@: k and 11 h-kj I E 0 implies h ~ k. h Next, if h = k then I!h-k! I E 0 and if in addition h g k then h ~ k by the first part of the theorem. Finally, we use Cauchy's formula to show that if h - "± k then h(z) ~ k(z) for z E ns(:'G). '~ all w E • bG and I When z E ns(:'G), 1 I z-w I is finite for 45 ih(z)-k(z) I ~ J ! hj:~~tll l<lwl + T (5. 3) J I:~=~: l<lwl ""0 . *bG\T THEOREM: = µ( r ), that is ~ determines r, the finest topology which agrees with s on bounded sets. Moreover, if h f k and µD(J') PROOF: b The first part is proved the same way that µD( =) = µ( S) was proved, namely by showing µD( ~) = inf(F n) where F n is the filter w n [{h, k) : 11h-k11 :5: n}' w E s. generated by the sets The second statement follows from the last theorem since if h r k then h ~ k and since l lh-k!I E 0, (5. 4) h g k. THEOREM: The strict t opology S(H(G), P(G)) is strictly coarser than i; which is coarser than r which in turn is coarser than the Mackey topology m(H( G), P( G)). In particular, S ~ m. PROOF: We prove S is strictly coarser than not coarser than [3. s by showing that a is Recall the example given above of an infinite power of the Riemann mapping of the inside of one boundary component of G. That function is a S-infin itesimal with modulus one on one whole boundary component, hence not an a-infinitesimal. (PA., O) E µ( S) and ( Thus p\ 0) r/: µ(a) n µ( S) and ~ is strictly finer than S. 46 r is finer than s by definition. a is coarser than the Mackey topology since any a-continuous functional has a representation as an integral: cp(h) = oo g EL f h(z)g(z) Idz j, bG (bG) and hence we may view a as a polar topology on a restricted collection of subsets of P(G) making a coarser than m. This means that over, s is coarser than m since both a and \3 are. More- s and \3 have the same closed convex sets since they are both compatible with the duality (H, P ). Finally, we show that r is compatible with the duality (H, P) and hence coarser than m. H(G).. Let L be a r-closed convex subset of We shall show that L is $-closed which implies the same continuous linear functionals. r and \3 have By the characterization of \3 as the topology of bounded sequential convergence it is enough to show that if (h : n EN) c L and h n - n B? h, then h E L. Since (h : n EN) is n bounded the s-closed convex hull and the r-cl osed convex hull coincide. Moreover, the r-closed convex hull is contained in L. Now since s and \3 have the same closed convex sets, having the same dual spaces, the \3-closed convex hull equals the r-closed convex hull and h E L. (5. 5) This completes the proof of the theorem. THEOREM:. h g. k implies h ~ k, that is the a-uniformity is finer than uni- form convergence on compact subsets of G. PROOF: I Take z E ns(:~G) so that (1 I z-w I ::; ME 0, for all w E >:~bG. the Cauchy formula h(z) ~ k(z), and the proof is finished. By 47 BIBLIOGRAPHY [l J N. Bourbaki, Elements of Mathematics: General Topology, English translation, Addison Wesley, Reading, 1966. [2 J W. A. J. Luxemburg, A General Theory of Monads, in the volume Applications of Model Theory to Algebra, Analysis, and Probability, Ed. W. A. J. Luxemburg, Holt, Rinehart and Winston, New York, 1969. [3 J M. Machover and J. Hirschfeld, Lectures on Non-Standard Analysis, Springer-Verlag lecture notes 94, Berlin, 1969. [4] A. Robinson, Non-Standard Analysis, Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam, 1966. [5] L. A. Rubel, Bounded convergence of analytic functions, BAMS 77 (1971), 13-24. [6] L. A. Rubel and J. V. Ryff, The bounded weak-star topology and the bounded analytic functions, J. Functional Analysis 5 (1970), 167-183. [7] L. A. Rubel and A. L. Shields, The space of bounded analytic functions on a region, Ann. Inst. Fourier (Grenoble) 16 {1966), fasc. 1, 235-277. [8 J K. D. Stroyan, Additional remarks on the theory of monads, presented at the 1970 Oberwolfach symposium on nonstandard analysis, to appear. 48 APPENDIX 1 THE NONSTANDARD HULL .AJ.'JD DUALITY Robinson [ 4] constructed the completion of a metric space by a nonstandard construction based on the extended metric. Luxemburg [2, Theorem 3. 15. 1 J extended Robinson's construction by showing that it applied to standard uniform spaces and that (in saturated ultrapower models) it gives rise to a larger complete space, standard hull, in which the completion is contained.. the~- Also, Machover and Hirschfeld [3] later extended Robinson's construction to standard uniform spaces using entourages. The ffrst result of this appendi x shows that Luxemburg's constructi on applies to internal or ~:~-uniform spaces. The motivation for this is not abstract generality, but rather to obtain H(D) from hull (H(G)) where G approximates >:~D as in section II. 3 above. are sure there are other applications as well. We The reader should consult Luxemburg's paper [2] for details not contained in this appendix. We remark that in [8] the author showed that the nonstandard hull of a standard precompact uniform space is complete for nonstandard models which are only enlargements. seems essential in the general case. The use of saturation However, we remind the reader that 6-incomplete ultrapowers (the simplest of all nonstandard models!) r are f\! 1 -saturated (Luxemberg [2, Theorem 1. 6. 4 ]). The outcome of this l.ast rema·rk is that nonstandard hulls of >!~-metric spaces are complete. 49 (Al. l) CONSTRUCTION OF THE 1 fr.JTERNAL 1 NONSTANDARD HULL. Let X be an internal set and A = [d: d EA} a (possibly exter- nal) set of >!'-pseudometrics on X, that is, each d EA is a map d: X XX --. >!<R + satisfying d(x, x) = 0, d(x, y) = d(y, x), and d(x, z) ~ d(x, y) + d( y, z). Let 0 A (a) be the A-galaxy of a, that is, the set of points a finite distance from a, [x EX: for each d EA, d(x, a) E O}. oA(x) = [y EX: d(x, y) ~ 0 for each d EA}. Let On the set OA(a) / oA' the points of 0 A (a) which have been identified if they are infinitesimally close for each d EA, the mappings st(d(x, y)) are pseudometrics. We denote the resulting uniform space by hull(X, a;A) , which we refer to as the nonstandard hull of X at a with respect to A. In the case of H(G) in section II, we take a to be the zero function and A the single element 11x-y11. (Al. 2) THEOREM: Let X be a >!<-uniform space in a nonstandard x-saturated ultra- power model >!<JC. X. Let a EX and J\ be a family of >!<-pseudometrics on If x > max(D[ , card (A)), then hull(X, a;A) is a complete uniform 0 space. The proof in Luxemburg [2, pp. 80 and 81 J applies with more work to extend the Cauchy set. We omit the details. The simpler case of the hull of a >!<-metric space is all we shall use and this is 50 done as follows. Let ( o(a ) : p EN) be a Cauchy sequence in hull(X, a; [d}). Extend a: ~:'N ~ ~:)C to an internal sequence. a -1 p A (X) 2 N we know a with w E #N. -1 (X) contains an initial segment [l,. Since e. , w} o(a ) is the limit . .W Now we shall consider the hulls of a ~:~-Banach space E and its internal continuous dual E'. We assume throughout"that we are working in a saturated ultra power model. and (E') 0 Let E 0 = hull (E, O; [ 11x-y11 }) = hull (E' , 0; [II cp- V11 }) • We may also consider the external continuous dual of the ordinary Banach space E (Al. 3) 0 which we .denote by (E 0 )'. THEOREM: (E')o is a closed subspace of (E )' which separates points of E • 0 0 PROOF: If Cf) EE' has finite norm, then stcp(x), xEE linear, and 11 st <P II ~ 11 cp I!. 0 is well defined, Now if 11 ~-cp II ~ 0, then st(~ (x)) = st(cp(x)) for x E E , so we may view (E') as norm embedded in (E )'. 0 0 0 It is closed since it is complete. If 0 -:/; x, y EE with 11x-y11 ~ 0 so that they give rise to distinct elements of E , we define <P as follows~ 0 cp(x/llx!I) = 1, cp(y/!IY!!) = 0 and Cf)([a/llxllJx+ [b/llYll ]y) = a and then extend cp internally to all E subject to the condition that 11cp11 :::= 1. Now st( co) separates x and y in E . 0 This last result leads us to ask whether (E' ) 0 is all of (E )' 0 and on the basis of the compatibility of nonstandard models with 51 linked algebraic and topological properties, we might expect this to be the case. On the other hand, the construction of nonstandard hulls is external and since E 0 may be quite large it also seems possible that (E )' may be strictly larger. 0 question. We do not know the answer to this 52 APPENDIX 2 SWEEPING MEASURES TO THE BOUNDARY OF SMOOTH REGIONS In this appendix we extend a result of Rubel and Shields from the unit disk to finitely connected regions with smooth boundary. The ~:~-transform of this result is used in section II. 3 on the approximating region to represent the pre-dual as internal integrals infinitesimally near the boundary of an arbitrary region. Since the techniques of Rubel and Shields [7] are scarcely changed we ·have put the result in an appendix. (A2. 1) THEOREM: Let G be a finitely connected region with smooth boundary bG. 1 1 Then M(G) /N(G) and L (bG) /N (bG) are isomorphic pre-dual Banach spaces for H(G). 1 Specifically, for µ E M(G) there is an f E L (bG) such that r hdµ = J f(z)h(z) Idz I for all h E H(G). Moreover' the norm in the G bG quotient spaces is preserved under this assignment and the assignment is onto. We begin by stating the lemmas we shall need in order to apply the techniques of Rubel and Shields. LEMMA 1: Each h E H(G) determines a bounded measurable function on bG by the non-tangential limits of h. holds on bG: Moreover, the C a uchy formula 53 h(z) = (1 /2 TTi} J (h(w) /(w-z) }dw , z EG • bG The smooth simply connected case follows from the fact that the inverse of the Riemann mapping is angle preserving to the boundary and absolutely continuous on the circle. The finitely con- nected case can be reduced to this by dividing the region with nonoverlapping smooth simply connected curves tangent to the successive boundary components. In this way you reduce the region to the study of two simply connected ones which overlap on the interiors of the dividing curves. The boundary integral 1 r = s+ J J bG c1 n 6 i=O c2 J r.1 and the Cauchy formula follows. LEMMA 2: 00 H (G} viewed as the non-tangential limit functions on bG is a 00 00 closed subspace of L (bG), in fact, the L -norm of the non-tangential 00 limit function equals the H -norm of the analytic function. This follows from the simply connected case by the same procedure as in Lemma 1. The simply connected case follows from the fact that sets of measure zero correspond under the Riemann mapping. 1 1 The importance of Lemma 2 is that L (bG} /N {bG} is a pre 00 dual for H (G). 54 LEMMA 3: 1 In the weak-star topolog ies cr(H, M) and cr(H, L ) a linear space · is closed if and only if it is sequentially closed. This well-known result appears in Banach's ·treatise. LEMMA 4: A sequence (hn) S. H(G) converges in cr(H, M) to a limit h if and only if (h ) converges boundedly to h. n This result appears in the paper of Rubel and Shields [7 ]. (See section II. 4 for the definition of bounded convergence.) PROOF OF THE THEOREM: 1 Suppose first that we are given a function f E L (bG). that L(h) = f f(z)h(z) !dz! is a-continuous by showing that its kernel bG is a-sequentially closed. (Lemma 3.) Assume h ~ h and L(h ) n L(h) = 0. assume We show Now h ~l. n n = 0 for each n. ~h by Lemma 4 so !!hn II : ; ; B and for convenience By Alaoglu's theorem h subsequence k n We wish to show ~ k. 1 for each f E L (bG). 1 1 has a a(H , L /N )-convergent CX) n This means By the Cauchy formula, substituting = f(z), we see that k(z) = h(z). z dz z::w ld.Zf Since L is by definition cr(H 00 , 1 1 L /N )- 1 1 continuous L(k) = 0 and therefore L(h) = 0 which shows L /N is contained in M/N. 55 Conversely, if K(h} = that the kernel of K is a(H and 3.} So we assume h n 00 , J h(z)dµ(z) for µ E M(G) we must show 1 1 L /N )- sequentially closed. o-(Hro, Ll / Nl) (Lemma 2 h, with K(h ) = 0 for all n. n We must show K(h) = O. By the uniform boundedness principle the h essentially bounded on bG and hence within G. wise by the Cauchy formula. K(h) = O. n are uniformly They converge point- By Lemma 4 therefore h ~h and n 56 APPENDIX 3 NONSTANDARD MODELS In abstract algebra one often studies the properties of a map- ping necessary to preserve the structure in question, that is, monomorphisms. The image under such a map is indistinguishable from the original for the sake of the structure in question. (The reals embedded in the complex plane, for example.) We wish to inject "ordinary mathematics'' into a larger theory in a way which preserves its structure, much as the real numbers are embedded in the complex numbers. The reader is no doubt familiar with examples of properties of the real numbers which can be more easily obtained by first embedding R in C. This is analogous to the way one may view applications of nonstandard analysis--we enlarge to simplify. For our purposes "ordinary mathematics" is the study. of the following set 'Y} called the superstructure based on the natural numbers, N. 'J = Let N 0 = N, Nk = co U Nk. Jk:;} N 1), k = 1,2, ... , we take ... ~ 1=0 (P denotes power set, the set of all subsets.) k=O The structure we are interested in is the membership relation, e, restricted to7( No doubt the reader "believes" that we may base mathematics on set theory--calling ~ a (standard) model for "ordinary mathematics" is no more than that ''belief". This set is sufficiently large to suit our purposes, for example, the real numbers are in as Dedekind cuts of rationals, which are ordered pairs of integers, which are ordered pairs of natural numbers. (We view (a, b) as 57 00 (a , [a, b} }. ) Functions are sets of ordered pairs, so H (D) E ?'/. Riemann surfaces can be embedded in Euclidean space, so they too can be viewed in 7/. (The reader can convince himself of these statements with a little effort.) Now we give the essential properties of a nonstandard extension map, >:', which is defined on 'I/. ,,, .,. . is an injection, it is one-to-one. >:' pre serves E: if a EA E /7, then >:ca E >:'A. >~ preserves equality: >:' [(x, x): x EA E'r/} = [(y, y): y E >:cA}~ ,t, ··· preserves finite sets: if a , a , .•• , an E 'lj, then 1 2 >:c ta f } r>!< >:~ } , ••. ,a n . t a , .•• , a n . 1 1 >:c preserves basic set operations: >:c¢ ¢, >:<(A U B) = = = >!<A U >!<B, >:c preserves domains and ranges of n-ary relations and commutes with permutations of n-ary relations. This allows us to extend functions to the extension of their domains and ranges for example. ''c ·· pre serves atomic standard definitions of sets: >:c [ ( x, y) : x E y E A E J? } = [(z, w) : z E w E >:'A } • >!c produces a proper extension, that is a NONstandard model: if A E 'l[ is an infinite set, then C'x: x E A} which is denoted properly contained in >:cA. A, is This means >:cR contains infinitesimals, but we remind the reader that it also means that the set of standard ~ ~ ~ subsets of ···R, P(R) is a proper subset of the internal subsets of '"R, The image of >!' is a nonstandard model of ordinary ma thematics in the sense that a theorem phrased in terms of sets of 'Y( is 58 true if and only if its >:~-transform is true, where the >:'-transform is the sarne statement with a >:' on each constant. For example, the nonstandard reals, >:'R, are >:'-complete and yet externally incomplete. This means every bounded set A E >!'P(R) has a least upper bound, nonetheless, the set of infinitesimals o E P('R) does not have a supremum. In particular, the set of internal subsets of >!'R, >!'P(R), is properly contained in the set of (all or) external subsets, P(:'R). Dis- tinguishing between internal and external objects is what allows us to "f/ and still preserve all of its properties so far as they can be expressed in terms of the extensions of sets in 'Y/. enlarge The >!~-transform of a statement from ordinary mathematics is the internal notion one obtains in the nonstandard model by first writing the statement in terms of E and sets . of'}(, being careful to only quantify over elements of~, stant in the sentence. and satisfies the and then placing a >:' on each con- A >:'-continuous function is necessarily internal ~·, B- o definition where B and 6 range over 'R. S-continuous function may be external and it satisfies the with standard € and E;- An o definition o, that is e, o ER+ --being able to work with both internal and external concepts like these makes the theory useful. It is usually difficult to tell whether a given set is internal, but . one case is easy--when the set is described by a sentence which involves only internal constants. We call this the IN"TERNAL DEFIN"I- TION PRIN"CIPLE and use it extensively above. A special case of this is when the sentence is the >:'-transform of a sentence from 6YJ, then the set is the image under >!' of the corresp ondingly defined set 59 One way to exhibit ~:<-mappings is by means of the ultrapower construction. Nonstandard ultrapower models have extension proper- ties for mappings as well as additional saturation, most important for applications, they will seem quite concrete to many mathematicians while an appeal to the compactness principle may not. We shall not give this construction since it can be found in many places. In order to work more freely with internal sets we assume our nonstandard model is x-saturated up to some infinite cardinal, usually card(~)+. Many interesting properties of x-saturated ultrapower models can be found in the paper, A General Theory of Monads by W . . A. Luxemburg. The above axioms for the >:<-mapping appear in A Set- Theoretical Characterization of Enlargements by A. Robinson and E. Zakon. J~ Both papers are in the volume Applications of Model Theory to Algebra, Analysis and Probability, edited by W. A. J. Luxemburg. For our purposes the properties of x-saturated ultrapower needed are Luxemburg's Theorems 2. 7. 11 and 2. 7. 12.