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M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces. 1. Preliminaries. In order to make this paper self-contained we include the definition of the two types of derivative used and the definition of an 8-category. We will employ the definitions of the derivative in topological linear spaces investigated in detail by Averbukh and Smolyanov [2], [31. See also [14], [15] and [191. Let TLS denote the class of all Hausdorff topological linear spaces over the real field R. Let 2 1(E, F) = 2(E, F) denote the set of all continuous linear maps from E into F, where E, F e TLS. We define by induction p(E, F) = 2(E, p 1(E, F)). Each ?1p(E, F) is given the topology of uniform convergence on bounded subsets of E. If X is a topological space, W(X) will denote the class of all open subsets of X. Let f: U -' V, where U e ((E), V E C(F), E e TLS and F c TLS. Then we say f is Fre6cbet differentiable at x e U, if there exists u e S(E, F) such that for each bounded subset B of E and for each o-neighbourhood W in F, there exists 8 > 0 such that f(x + th) f(x) u * th E tW, whenever b E B and ItL

This paper constructs a basis for C([O, 1]) which converges weakly to zero whose elements are nevertheless norm bounded away from zero. Introduction. A basis for a Banach space X is a sequence {x, } CX such that to each yEzX there exists a unique sequence of scalars cn for which the partial sums of E CnXn converge to y in the norm of X. Bases can be distinguished by combinatorial and topological criteria (cf. [4]). Foiaa and Singer asked in [1] if C([0, 1]) possesses a basis of type wco, that is, a basis that converges weakly to zero whose elements have norms bounded away from zero. This paper constructs such a basis using a scheme that may be useful in looking for other peculiar bases in C( [0, 1]). J. R. Holub has recently [2] constructed a basis in co that is weakly closed yet intersects every weak neighborhood of zero defined by a single linear functional. The referee has pointed out that Holub's example and the construction below easily imply the existence of such a basis in C( [0, 1]). A method of construction. A sequence of piecewise linear functions which form a basis in C([0, 1]) can be constructed in the following way. The sequence of functions will be accompanied by a parallel sequence of points in [0, 1]. These points will be called joints. The joints are to mark the intervals on which the functions are actually linear. The functions and joints will be grouped into successive stages, each stage having some finite number of functions and an equal number of joints. The kth function (joint) of the nth stage will be denoted fn,k (an,k). To simplify notation an,k will sometimes be written a(n, k). To form a basis thef.,k must be linearly ordered. Do this by sayingfi,j precedesfn,k if i

Introduction. In a recent paper, [l], A. Beurling has shown that the positive translates of an integrable function defined on [0, m0) generate, in a certain sense, at least one exponential of the form e-iz, x > O, Ia < 0, provided that the function does not vanish outside a finite interval. It is the converse problem with which we shall be concerned here; namely, to what extent can the exponentials so generated be used to approximate the given function. We are able to give what amounts to a complete solution. The situation resembles strongly that of Schwartz' theory of mean periodic functions [2]. M. Kahane has shown in [3] (see also [41) how this theory can be presented very simply using the notion of Fourier transform of a mean periodic function. Beurling also made use of this method in the present case; however, we shall find it convenient to exploit this tool more systematically, in closer analogy with Kahane's work. We shall also study our approximations in a topology (the same as the one used in the theory of mean periodic functions) which is simpler than that of Beurling. Beurling based his work mainly on a certain division theorem which states roughly that an entire function is of finite exponential type if it is bounded on a half plane and equal to the ratio of two bounded analytic functions on the complementary half plane. The conclusions we make here follow from a refinement of this given in ?3 which yields an upper bound for the type of such an entire function. It should be remarked that B. Nyman ([7, pp. 28-29]) has established a result similar to the one given here, using, however, a quite different topology. (The referee calls attention to this in his report; although I have since had the opportunity to consult Nyman's work, it was not accessible to me in New York at the first writing of this paper.)

Seemingly disparate comparison theorems for nonselfadjoint elliptic equations have been established by C. A. Swanson and the author. The present paper establishes a class of comparison theorems involving an arbitrary vector valued function p(x). Special choices of p(x) yield both of the comparison theorems cited above. Several papers have recently concerned themselves with generalizations of Sturm's comparison theorem to nonselfadjoint elliptic equations. The first such theorem was due to Protter [1], and subsequent generalizations have been established by Swanson [2 ] and the author [3 ]. An interesting aspect of [2 ] and [3 ] is the seemingly disparate hypotheses which the principal theorems require. The purpose of this paper is to establish a class of comparison theorems for nonselfadjoint elliptic equations which contain the results of [2] and [3] as special cases. Our method consists of combining appropriate generalizations of techniques used by Picard [4] and Picone [5]. Let D denote a smooth bounded domain in Rn and let u(x) and v(x) be nontrivial real valued solutions of the elliptic equations ) a Eij(x)-+ 2 Z bi(x) + c(x)u = 0, -i=1 aXi axi i(l (x) (2) -E aij(x)+ 2 E Zfl(x) -+ y(x)v = 0. i, j= i 0xj d9xi i=1 d9xi It is assumed that the coefficients a,j(x), aij(x), bi(x) and fi(x) are of class C' in D while c(x) and y(x) are to be continuous in D7 (some of these requirements can be weakened somewhat as will be clear from the manipulations below). Furthermore (1) and (2) are to be uniformly elliptic in D. Received by the editors September 4, 1970 and, in revised form, November 16, 1970. A MS 1969 subject classifications. Primary 3511.

In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map. I. Preliminaries. In this paper all spaces will be simplicial complexes and all maps will be piecewise linear. The boundary, closure, and interior of a subset X of a space Y will be denoted by bd (X), cl (X) and int (X), respectively. A proper map f, taking a space X into a space Y, is a map such that f bd (X) = bd (Y) n f(X). A Dehn map f, taking a compact, bounded, two-manifold F into a 3-manifold M, is a map such that (1) bd (F)=f-lfbd (F); (2) fIbd (F) is a homeomorphism. All unlabeled homomorphisms are natural maps induced by inclusion. We shall consider the problem of replacing a proper Dehn map of a compact, connected, orientable, bounded surface F into a compact, (necessarily) bounded, irreducible, orientable 3-manifold M with an embedding. As will be pointed out later, the requirement of orientability may not be strictly necessary. The author would like to thank J. Gross for a number of helpful conversations. In [3] Papakyriakopoulos proved Dehn's lemma, that is, a proper Dehn map f of a disk 9 into a 3-manifold M can be replaced by a proper embedding g of 9 into M such that g bd (9) =f bd (9). A natural question to consider is the following: Let f: F -* M be a proper Dehn map of a compact, connected, bounded, orientable surface F into a compact 3-manifold M. Does there always exist a proper embedding g: F1 -? M such that (1) g(bd (F1))'f(bd (F)); (2) genus (F1)

Introduction. The concept of isotopy, recently introduced(') by A. A. Albert in connection with the theory of linear non-associative algebras, appears to have its value in the theory of quasigroups. Conversely, the author has been able to use quasigroups(2) in the study of linear non-associative algebras. The present paper is primarily intended as an illustration of the usefulness of isotopy in quasigroup-theory and as groundwork for a later paper on algebras, but is bounded by neither of these aspects. The first two sections are devoted to the basic definitions of quasigroup and isotopy, along with some elementary remarks and two fundamental theorems due to Albert. Then there is initiated a study of special types of quasigroup, beginning with quasigroups with the inverse property (I. P. quasigroups). A system Q of elements a, b, * * * is called an I. P. quasigroup if it possesses a single-valued binary operation ab and there exist two one-to-one reversible mappings L and R of Q on itself such that

This paper gives a necessary and sufficient condition in order that a bounded linear mapping from Ll(u) into a Banach space be compact. It is applied to provide a slightly improved form of the Radon-Nikoydm theorem for vector valued measures and to give a sufficient condition in order that the range of a vector valued measure of bounded variation be compact. In this paper, we give a sufficient condition in order that a measure is representable as an indefinite Bochner integral. We will show that a bounded linear operator Tmapping L'(,u) into a Banach space is compact if the set {T(pM1/1iz(Mi)): Mi E X} is precompact for every sequence of disjoint measurable sets {Mi} (see Theorem 1). Combining this with a result of Rieffel [2] gives a slightly improved Radon-Nikodym theorem for the Bochner integral. We apply this result in Theorem 2 to give a sufficient condition in order that the range of a vector valued measure be precompact. Here, we follow the approach of Uhl [3]. LEMMA. Let (X, X, j) be a finite measure space. Let T: Ll(,u)-*F be a bounded linear operator, where F is a Banach space. For each positive real number c, define R(c)={T(pM/,u(M)):M E X,O

Introduction. Let D be an arbitrary domain on the Riemann sphere (that is, D is an open connected set on the sphere) and let p be a point of D. Let H '(D) denote the space of bounded analytic functions on D. We consider the problem of maximizing If'(p)l for fin H-(D) with If(z)lj< 1 for all z in D, and f(p) =0. In the case when D is the unit disc U and p =0, this is Schwarz's lemma and the solution is classical: the maximum is one and if If'(0) = 1, then f(z) = eiOz for some real constant 0. In the general case, the theory of normal families assures us that there is at least one fo in H (D) with foll? 1 I and fo(p) ? f'(p)I for all f in H (D) bounded by 1 which vanish at p. We call any such function extremal. Theorem 1 of this paper provides an elementary proof that there is only one extremal function. We thus add to our problem the question of investigating the properties of the extremal function. In the case when D is bounded by a finite number of disjoint analytic simple closed curves, the properties of the extremal function have been studied extensively (see [1] and [7]). In this case, the extremal function is known to be analytic over the boundary and to have modulus one there. We show that this is a local property and carries over to all points in the boundary of D where the boundary is sufficiently well behaved, no matter what the remainder of the boundary of D is like. In ?1 we also show that the extremal function is closely related to the problem of removable singularities for bounded analytic functions. In ?2 we discuss a related extremal problem and show that the properties of its solution imply that on many planar domains D, each function analytic and bounded by 1 on D may be approximated uniformly on compact subsets of D by a sequence of inner functions. We also briefly discuss extensions of the results to open Riemann surfaces. Finally, some remarks on previous work on Schwarz's lemma are in order. Carleson [3, pp. 78-82] and Havinson [8, Theorem 9] have previously shown that the extremal function is unique. Our proof differs substantially from each of theirs and is considerably more elementary. Furthermore, it does not make use of any of the results from the finitely-connected case. Also, our Theorems 2 and 3 and a weaker version of Theorem 4 have previously been proved by Havinson [8, Theorems 22, 28, and 20]; however, as in the case of Theorem 1, the proofs given here are appreciably simpler and more elementary. Indeed, part of the motivation for this paper was to find simpler and more transparent proofs of these results. In particular, it is only in the proof of Theorem 5, which describes the behavior

In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zero-free analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed. Let C denote the unit circumference in the z-plane and D its interior. The term C-polynomial shall mean a polynomial all of whose zeros lie on C. In [6] the first author unified the results of MacLane [5] and Chui [2 ] by proving THEOREM A. Let f(z) = 1 +c+c2z 2+ * * * be a zero-free holomorphic function in D. Then there exists a sequence of C-polynomials Pn(z) assuming the value one at z =0 which converges to f(z) uniformly on every closed subset of D. If in addition M-supizi

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].

Introduction. In the preceding paper [8] we proved that if all harmonic functions in n-dimensional bounded domain D satisfy the mean value property (MVP) with respect to some point P and a given density function ,t (volumedensity, surface-density, etc.) then, under some simple assumptions on ,, D must necessarily be a ball with center P. In the present paper we are interested in studying the MVP from a complementary point of view, namely, we are interested in finding conditions on ,u (u>?0) under which the MVP holds at most for a finite number of linearly independent functions. The MVP is meant to be

Introduction. Problems of great interest in real-variable theory are the analysis of the structure of a function whose derivative is finite, and the determination of a function when its derivative is given and is finite at each point. The study of these problems has led to a theory of non-absolutely convergent integrals.t The corresponding problems when the derivative in question is with respect to a function of bounded variation have received little attention, and it is with these that the present paper is mainly concerned. In this case we find that there is involved a theory of non-absolutely convergent integrals with respect to a function of bounded variation. Lebesguet has given these questions some consideration. His results depend, in part, on a transformation which changes integration with respect to a function into ordinary integration. By means of this transformation very general results are easily obtained. But for dealing with some particular situations it is not altogether suitable; for example, a discussion of integral equations which involve integration with respect to a function of bounded variation. For this reason we have, in the present work, adhered to methods which are direct. Incidental to our main purpose is a study of the derivative of a function F(x) with respect to a function of bounded variation a(x). Here, too, we have been preceded by others, chiefly Daniell.? In the Transactions paper Daniell concerns himself with the central derivative with respect to a

Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then

The importance and usefulness of Taylor's theorem need not be dwelt upon here. We are interested in it for functions whose arguments and functional values belong to abstract spaces of the Frechet type. Consequently no Rolle's theorem can be even stated, and the proofs will be somewhat different from the usual proofs in the theory of numerically-valued functions. It is also to be expected that a slight strengthening of hypotheses will be required. However, it is not necessary to assume the existence of a uniformly continuous nth differential, as was done in the first announcement of these results.: It is sufficient that the function should have an nth variation (in the sense of Gateaux), with certain limitations on its discontinuities. The functions we discuss will be one-valued functions whose arguments x and functional values y belong to linear metric spaces X and ) respectively, briefly, functions F on X to S9. Linear metric spaces correspond to the spaces called by Frechet, "espaces (D) vectoriels." For the notations, postulates and fundamental propositions, we refer the reader to Part 1 of a paper by T. H. Hildebrandt and the author,? to avoid practically entire repetition of that section. No other parts of that paper are needed here. We shall consistently use the letters r, s, a, b, c to refer to real numbers, and the German t to refer to the real axis. The notation To will then refer to a set of open intervals of that axis. By the notation (ab) we shall mean the bounded closed interval of St with end points a and b. Whenever the space M) is required to be complete, this will be specifically mentioned. The form of remainder obtained in our Taylor's formula is a generalization of that given by Jordan|| and analogous at least to those obtained in

Introduction. Let G be a locally compact topological group which is unimodular, that is, the left and right Haar measures coincide. We propose, in this and subsequent papers, to investigate the structure of its "L2-system." By the "L2-system" we mean the (complex) Hilbert space formed from the complex-valued measurable functions f on G for which If| 2 is integrable with respect to the Haar measure, and with a product defined in terms of convolution. This product is not always defined (for that reason we call it an L2-system rather than an L2-algebra) but there are dense subsets (for example, L,nL2) on which multiplication always is defined so it is "almost" an algebra. Our theorems depend only on certain key properties of these L2-systems so we shall investigate general systems with these properties, calling them H-systems. We do this not only for generality but because when decomposing an L2-system into minimal parts we expect those parts to be H-systems though they need not be the L2-system of any group. This paper contains the definitions of an H-system and an L2-system; the proof that an L2-system is an H-system; theorems on idempotents and generation of ideals by idempotents; a functional calculus of self-adjoint elements in an H-system; a simple structure theorem for abelian H-systems; and an example of the L2-system of a special group, which has an interesting structure. Using the theorems of this paper we believe we shall be able to obtain the complete structure of H-systems--the ultimate theorems presumably asserting that every such system is a "direct integral" of simple systems, and that every simple system is a certain type of full matrix systembut with continuous rather than discrete matrices. At present these theorems are not proved but before even considering them carefully it seems necessary to develop the material of this paper as a foundation. One might ask why we choose to investigate this L2-system, in which multiplication is only partially defined, when there are other systems (for example, L1-the integrable functions) which are actually algebras and whose structure is still unknown. Our belief is that in spite of having only a partially defined multiplication our system may be easier to study, in somewhat the same way that certain questions are easier to answer about unbounded hypermaximal operators on Hilbert space than about bounded operators on a general Banach space. That there is a relation between that situation and ours is shown by the fact that left (or right) multiplication by a fixed element in an L2-system is, for many choices of the element, an unbounded hyper

as well as certain extensions of the well-known maximuim principle. In the usual type of boundary value problem, the domain of the solution lies in the upper half plane t >0 and part of its boundary lies on the hyperplane t = 0. The data prescribed on the latter part are referred to as initial data. The present paper, on the other hand, treats domains extending indefinitely in the t-* co direction. Thus instead of initial-value problems, we consider "generalized steady-state problems" (see [2]) or "problems without initial conditions." A particular case of such a domain is the cylinder QX(oo, m), where Q is a bounded domain in x-space. Some existence and uniqueness questions for boundary value problems in such a domain were treated in a previous paper [2] by the author. Another special case is the complement of such a cylindrical domain; here some obvious results can be obtained by combining the maximum principle developed in our Theorem 4 with the technique of Meyers and Serrin [4]. (See also [3] for certain results in one space dimension when the solution is periodic in time.) In the present paper, however, the domain of the solution (which we hereafter denote by c) is generally allowed to be noncylindrical; the goal in fact is to obtain geometric properties of D which guarantee uniqueness within the class of bounded solutions. We shall usually require no regularity properties of the coefficients of L other than local boundedness, and for our general result (Theorem 3) we assume only that L is locally parabolic. We shall be concerned with functions satisfying Lu >0. The well-known standard weak maximum principle holds for such functions if the derivatives appearing in the operator are continuous; we shall designate these functions by the term subsolutions.

without the assumption of the continuity of the derivative f'(Z) on the closed region R bounded by the curve of integration C, and thereby he has laid deeper foundations for the CAUCHY-RIEMANN theory of fuinctions of the complex variable. An abstract of these memoirs is to be found in the Bulletin of this Society for June, 1899, pp. 427-429. GOURSAT set out by a direct process t to evaluate the integral in question. In the present paper, by an indirect process, I prove that the integral has the value 0. The essential elements of the proof are those of GOURSAT'S first paper; by the modification indicated, and by the imposition on the curve C of a certain condition fulfilled by all the usual curves, one avoids the necessity of introducing the lenmma to which GOURSAT'S second paper is devoted. The necessary preliminary definitions and theorems are given in some detail in ? 1, in which connection I refer especially to JORDAN'S Cours d'Analyse, 2d ed., vol. 1, 1893, and to HURWITZ'S address at the Zurich Congress of 1897 entitled.: Uber die Entwickelung der (illyemeinen Theorie der analytischen Functionen in neuer er Zeit (Verhandlungen des iilathematiker Kongresses inZiirich . .; Teubner, 1898). Then in ?? 2 and 3 I state and prove the two

The matrix A may be chosen to be a Jacobi matrix, that is, Ajj=0 for i-jI >1; and An,n+l> 0 may be required. If I is {0, 1, 2, * }, A is then determined uniquely, assuming the convention that any invariant subspace of A orthogonal to the 0th coordinate subspace will be ignored. (Assuming, that is, that if Ak,k?l =Ak1, k = 0 then in A = ((A ij)) indices will be let run up to k only.) For finite I, A is not in general determined uniquely. This paper gives a convenient canonical form for A. There is little trouble in including in this result the generalized sort of moment problem introduced by Nagy [6], where un above are in 63, the set of bounded self-adjoint operators on a Hilbert space SC. One virtue of the generalized problem is its application to the classical problem; see Proposition 4 below. Accordingly some of the facts outlined in the preceding paragraph may as well be proved for the generalized problem; this is done in ?2, which in fact is essentially a recitation in the wider setting of the proof for the classical case. The main result is in ?3. The following sections mostly examine the classical problem in light of it. I have found the canonical form handy for getting numerical bounds on momenits, but will not discuss this use further. Throughout the paper I use the notation a--1 -a, where a may be a number or an operator. If it is an operator this requires some understanding as to the space on which it operates, but I think I have avoided amlbiguity.

Introduction. Martingale theory for a linearly ordered index set in a finite measure space has been systematically discussed by Doob [4]. As convergence theory and system theory are the two main topics in martingale theory, Helms [6] has extended Doob's results on mean convergence to martingales in a finite measure space, indexed by directed sets, and Krickeberg [8; 9] extended the results of pointwise convergence and stochastic convergence to martingales in a a-finite measure space, indexed by directed sets. Krickeberg uses measure algebra and lattice theoretic methods, and in [8] makes the assumption in pointwise convergence that the martingales have some covering property, which is called Vo in his paper. Dieudonne [3] has given a martingale which is not pointwise convergent and has non-negative, bounded functions in a finite measure space, indexed by a countable directed set. Therefore, some additional conditions like Krickeberg's Vo to the usual ones seem to be necessary for pointwise convergence. However, the condition Vo is too weak to be the only additional condition needed for pointwise convergence of martingales indexed by noncountable directed sets. Some conditions such as terminal separability are needed. For the system theory, Bochner [2] has given some extensions to martingales indexed by directed sets, but without proofs. Some of his results are, unfortunately, not true. Snell [14] has defined regularity (which is closely related to system theorems) of martingales in a finite measure space, indexed by positive integers, and has given some sufficient conditions. He has not discussed the necessary conditions, nor the relation between regularity and the system theorems. In this paper, ?1 gives preliminary definitions and notation. ?2 is devoted to the definition of conditional expectation in a a-finite measure space W, and ?3 to that of martingales indexed by directed sets in W as well as to individual system theorems and inequalities. In ?4, a general pointwise convergence theorem is proved, and a new kind of convergence theorem, based on a differentiation theorem of interval functions [13, p. 192] is given. We

One of the themes of Edwin Hewitt's fundamental and stimulating work [16] is that the Q-spaces (now called realcompact spaces) introduced there, although they are not in general compact, enjoy many attributes similar to those possessed by compact spaces; and that the canonical realcompactification vX associated with a given completely regular Hausdorff space X bears much the same relation to the ring C(X) of real-valued continuous functions on X as does the Stone-Cech compactification gX to the ring C*(X) of bounded elements of C(X). Much of the Gillman-Jerison textbook [12] may be considered to be an amplification of this point. Over and over again the reader is treated to a "/" theorem and then, some pages later, to its "v" analogue. One of the most elegant "/3" theorems whose "v" analogue remains unproved and even unstated is the following, given by Irving Glicksberg in [13] and reproved later by another method in [11] by Zdenek Frolik: For infinite spaces X and Y, the relation 3(X x Y) = gX x / Y holds if and only if X x Y is pseudocompact. The present paper is an outgrowth of the author's unsuccessful attempt to characterize those pairs of spaces (X, Y) for which v(X x Y) = vX x v Y. It is shown (Theorem 2.4) that, barring the existence of measurable cardinals, the relation holds whenever Y is a k-space and vX is locally compact; and more generally (Theorem 4.5) that the relation holds whenever the k-space Y and the locally compact space vX admit no compact subsets of measurable cardinal. The question arises naturally as to when it will occur that vX is locally compact. Our best result in this direction, a part of Theorem 4.8, is, again, not definitive: X is locally pseudocompact if and only if there is a locally compact space Y for which Xc Yc vX. Some of the questions treated here are susceptible to attack when thrown into the uniform space context. See Onuchic [18] and, more intensively, Isbell [17, especially Chapter 8]. I am indebted to Tony Hager for several references, and for allowing me to announce here that, in addition to the results of [15], he has achieved certain new theorems on the relation v(X x Y) = vX x v Y, which being " uniform " and not "topological" have negligible overlap with those of this paper.

Preface. For some time it has been recognized that the properties of symmetry and bounded closure are of cardinal importance in the theory of locally convex spaces. The purpose of this paper is to discuss symmetry and bounded closure with the aid of a new topology, and to discuss the new topology itself. The introduction includes the preliminaries to the rest of the paper. The account of the basic theory of locally convex spaces is fairly complete, but brief. We have given proofs only when they represent some contribution. As the references indicate, most of the theorems of the introduction can be found in the literature. In ?1 we introduce the new topology, called the GF topology, and develop the theory of the GF space. A space is a GF space if and only if it is symmetric and boundedly closed. Study of the GF topology on duals leads to information about interior maps and quotient spaces, information which is not obvious from direct consideration of symmetry and bounded closure, We conclude the section with several examples. In view of the general results of the first section it is important to know what spaces are GF spaces. In ?2 we take up this question for direct products and their duals, and also the questions of symmetry and bounded closure. ?3 serves the same purpose for continuous function spaces. Finally in ?4 we discuss metric and LF spaces and their duals. The influence of the works of G. W. Mackey, J. Dieudonne, and L. Schwartz is apparent throughout the paper, and it is to these authors that we owe our interest in locally convex spaces. The terminology used in the paper is almost exclusively that employed in the works of Dieudonne, Schwartz, and Bourbaki. For the convenience of the reader we list the exceptions. We distinguish between weak and weak-star topologies; our total bounded is the French precompact and corresponds to the definition given in [14]; we owe the term relatively strong to Mackey [13]; where the French authors use filters we make use of nets, as in Kelley [10]; and for a uniform structure topology the natural definition of a Cauchy net leads to a concept of completeness which is equivalent to the French term and stronger than that given in [14]. There are no French equivalents for symmetry and bounded closure.

The first chapter of this paper concerns itself with questions of uniform boundedness of sets of points in a Banach space and sets of functionals on a Banach space, as well as with a group of closely related resonance theorems. A well known example coming under this heading is the theorem of Toeplitz [36] t stating that supm Z I= a,.I < oo providing 77m =Zn-, -amntn converges whenever S, does. Another is the theorem of Hahn [12 ] stating that if an arbitrary continuous function has the partial sums of its Fourier expansion, with respect to an orthonormal sequence of bounded functions (n, essentially bounded, then the sequence f ZX= 1c^,=(x)co,(t) I dt is also essentially bounded. Still another is the theorem stating that if the adjoint of an everywhere defined transformation between Banach spaces is everywhere defined, then the transformation is continuous. This was proved, at least for Hilbert space, by von Neumann [21 ], Stone [34 ], Tamarkin ( [34], p. iv), and Stone and Tamarkin [35 ] and is probably not usually thought of as a theorem on uniform boundedness. Most of the results w-e have in the first chapter are new, others have been proved only in special cases, and some are well known but the proofs heretofore given have been different. Previous methods for discussing questions of uniform boundedness divide themselves into three groups (i) those associated with the names of Lebesgue [18], Banach [2], Hahn [11], and Hildebrandt [13 ]; (ii) those characterized by the elegant and direct use of the Baire category theorem as in the works of Banach [1], Saks [31], Saks and Tamarkin [32], and others; and (iii) those employed in a recent theorem of Gelfand [9] the proof of which is closely related to that of the category theorem. The second chapter of this paper is concerned with cases where a weak limiting process implies a strong one. These cases seem to be rather rare but probably more will come to light in the future. The theorem quoted above, stating that the existence of the adjoint implies the continuity of the function, belongs to the class of questions discussed in Chapter II as well as those discussed in Chapter I, while its analogue (Theorem 42) in the Boolean ring of

Introduction. By "the space of continuous functioins" we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In ??1-3, we collect the properties of vector lattices which we need in the paper. In ?4 we add some properties of the first dual of C-the space of Radon measures on X-denoted by L in the present paper. In ?5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element f of M on an element ,i of L is denoted as usual by f(,u). If f lies in C then of course f(,u) =,u(f) (usually written f fd,u), and thus the multiplication between M and L is an extension of the multiplication between L and C. Now in integration theory, for each ,uEL, there is a standard "extension" of ffd,u to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called ,i-integrable. The question arises: how is this "extension" related to the multiplication between M and L. ??6-8 (and ?12) are devoted to this question. We show that given a bounded function f on X the equivalence class of M represented by f contains two distinguished elements, which we denote by f* and f*, such that for every Radon measure ,u, f*(,u) = f*fd,u and f*(,u) = f*fdM, where f* and f* are the lower and upper integrals of f with respect to ,u. Moreover, if f is integrable for every Radon measure, then f* and f* coincide, and we have a unique element whose value on each ,u is ffd4. In ?9 we study order-convergence in M. It is in this area that Af offers its most striking immediate gain in clarity over consideration of only the bounded functions oil X. We cite two examples. The first example: The characteristic functions of finite subsets of the real initerval 0< x < 1 form a directed system which order-converges to the constant function 1, while their Lebesgue integrals, being all zero, do not converge to the Lebesgue integral of 1. In M, however, this directed system order-converges to an element different from 1, and the above unsatisfactory situation disappears. In point of fact, order-convergence is perfectly well-behaved in M: if a directed system order converges, its values on every Radon measure converge

By using the results of [6], it is proved that for an extensive class of increasing functions h, ( ∗ ) ∑ 1 ≦ d ≦ x μ ( d ) d h ( x d ) ∼ x h ′ ( x ) as x → ∞ \begin{equation}\tag {$*$}\sum \limits _{1 \leqq d \leqq x} {\frac {{\mu (d)}}{d}h\left ( {\frac {x}{d}} \right )} \sim xh’(x)\quad {\text {as}}\;x \to \infty \end{equation} where μ \mu denotes the Möbius function. This result incidentally settles affirmatively Remark (iii) of [6], and refines the Tauberian Theorem 2 of that paper. It is also shown that one type of condition imposed in [6] is necessary to the truth of the cited Theorem 2, at least if some sort of quasi-Riemann hypothesis is true. Nevertheless, examples are given to show that on the one hand ( ∗ ) ( ^\ast ) may be true for functions not covered by the first theorem of this paper, and on the other that some sort of nonnaïve condition on a function h is necessary to ensure the truth of ( ∗ ) (^\ast ) .

1. A mean on a semi-group 2 is a positive linear functional of norm one on the space m(z) of bounded, real-valued functions on 1. A bounded semigroup S of linear operators from a Banach space B to itself is called ergodic if there exists a system cd of averages A such that for every S in s limA (AS-A)=limA (SA -A) =0; we have three strengths of ergodicity of S according as uniform, strong, or weak convergence is used in the operator algebra. The first part of this paper deals with the relationship between existence of invariant means and ergodicity of bounded representations. In Theorem 2 it is shown that weak ergodicity of every bounded representation of I is equivalent to weak ergodicity of the right and left representations of f by right and left translations on m(2), and equivalent to the existence of a mean on m(z) invariant under right and left translations. These conditions, in turn, are equivalent to existence of a directed system of finite means on mn(z) converging weakly to two-sided invariance under all right and left translations of m(z). Uniform ergodicity is similarly related to existence of finite means converging in the norm of m(z2)* to two-sided invariance (Theorem 4). The second part of this paper gives some sufficient conditions for existence of invariant means or of finite means converging in norm to invariance. For the former, the Markoff method of proof by fixed-point arguments is applied (?5) to "solvable" semi-groups and groups, and to semi-groups which are the union of expanding directed systems of sub-semi-groups with means. (For example, a group G such that every finite subset generates a finite subgroup has an invariant mean.) It is also proved by a direct construction (Theorem 6) that if G is a normal subgroup of a group H and if G and H/G have two-sided invariant means, so has H. In ?6 a parallel result is proved for finite means converging in norm to invariance. These results greatly increase the family of groups known to have invariant means. Solvable groups formed the only such class previously known; ?6 now shows that a solvable group satisfies a stronger property; it has a system of finite means converging in norm to invariance.

The object of this paper is to compare the summability methods of almost convergence and order summability. This is done by examining a method of summability which includes both methods. The main result is that every monotone Fourier effective matrix is strongly regular. Introduction. In this paper a general, nonmatrix method of summability is introduced which not only includes matrix summability but also includes the order summability of Jurkat and Peyerimhoff [2] and the method of almost convergence of G. G. Lorentz [5]. Some general theorems about this method are given and they are used to examine the relationship between the two methods mentioned above and the relationship between almost convergence and Fourier effectiveness. We are able to show that every monotone Fourier effective matrix is strongly regular. In ?1, we define the method and give some examples. The study of the mapping properties of this method comprise ?2. As applications of this method, comparison theorems between order summability and almost convergence are proved in ?3. 1. Definitions and examples. For v=1, 2, * * , let AV=(a'nk) be an infinite matrix of complex (or real) numbers. Let _' denote the sequence of matrices {Av}. For a sequence of complex (or real) numbers, x=fx,}, the double sequence t={tv}, defined by tv=k7 l av kXk is called the {Av} or S'-transform of x whenever the series converges for all n and v. A sequence x is said to be {Av} or Q'-summable to some number L if {tv} converges to L as n tends to oo uniformly for v= 1, 2, . L is said to be the S'-limit of x, written lim?, x=L or slx--L, and we say six is convergent to L. Also, let c,1 be the set of all sequences whose si-transform is convergent. v is said to be a regular sequence of matrices if x-*L implies six--L. We shall denote the set of null sequences by n and the bounded sequences by nn. Received by the editors April 25, 1972. AMS (MOS) subject classifications (1970). Primary 40C99, 40D05, 40G05; Secondary 40D25, 42A20.

The Fourier transform of an unbounded spectral distribution is studied: An explicit integral representation is obtained; connections are drawn to the associated generalized scalar operator. It is proved that every generalized pseudo-hermitian operator is the infinitesimal generator of a temperate CO group. Introduction. In [6] the author introduced a theory of unbounded spectral distributions in Banach spaces, and a corresponding theory of the generalized scalar operators which they represent. Properties of these objects were studied, culminating in a spectral mapping theorem [6, Theorem 4]. In this paper we study the Fourier transform of an unbounded spectral distribution, deriving an explicit integral representation, as well as growth estimates at infinity. The case of real support is considered in some detail, leading to the proof that every generalized pseudo-hermitian operator (i.e., generalized scalar with real spectrum) is the infinitesimal generator of a temperate group. This result generalizes the corresponding result for bounded operators proved in [5]. In this paper, all definitions are as in [6]. Integral representation of the Jourier transform. THEOREM I (EXTENSION OF SPECTRAL DISTRIBUTIONS). Let T be a spectral distribution [6, Definition 2] in a Banach space X. Then (a) for each fuinction ffromn the space (R2) of C' functions wt ith bounded lerivatives, and for each x E X, limit,, Tf pnx exists, where {c p} is a sequence of test flnctions as in [6, Definition 2(c)]. (b) If we define Tfx=Jimitn Tfp,,x for all x E X, then Tf is in the space Y(X) of all bounded linear operators on X, and the correspondence f--*Tf is a continuous linear mapping of X(R2) into f(X). (c) 7 T= Tf T, for al/f, g E .6(R2). (d) The operator Tf is independent of the sequence {p7J PROOF. (a) Define a double sequence of functions m-f(%-pm). Then {y form i X(R 2)-bounded subset of ?/ (R2) (cf. [3, p. 91 ] for the Received by the editors October 20, 1971. AAIS 1970 sub/jecr ch ssifications. Primary 47A65, 47A60, 47B40; Secondary 46F99. ? American Mathematical Society 1972

It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of R3 has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder's fixed point theorem to an appropriate mapping. Introduction. The purpose of this paper is to prove the existence and uniqueness of classical solutions to the mixed initial-boundary value problem for the Euler equations for an ideal incompressible fluid in bounded domains of R3 for a small time interval. This has been shown by Ebin and Marsden [3] for compact Riemannian manifolds possibly with boundary. In this paper we obtain a new proof by converting the equations to an equivalent system involving the vorticity and solving this by using the Schauder fixed point theorem. The technique is similar to that of Kato [4] where he proved existence of ideal flow in bounded domains of the plane for an arbitrary time interval. The author also used a similar method to obtain an existence result for ideal flow in all R3 by showing that a solution is the limit of Navier-Stokes flow as the viscosity goes to zero [7]. Existence of a solution for a short time in all R3 was originally shown by Lichtenstein [5]. The Euler equations for ideal incompressible flow in a domain D in R3 with boundary bD are (Et) ar/at + (r grad)r =-grad P + B, V * U = O, with constraints U n = 0 on bD and r(0) = A, where x is a point in D U bD; t E [O, T], Iv(X, t) is the velocity vector, P(x, t) is the (scalar) pressure, B(x, t) is the external force field vector, n(x) is the outward normal vector to bD and A(x) is the initial velocity vector. By formally computing the curl of (E') we get the system Presented to the Society, January 18, 1972 under the title The existence and uniqeness of nonstationary idealflow in bounded domains in R3; received by the editors October 12, 1971 and, in revised form, March 20, 1972. A J1S (MOS) subject classifications (1970). Primary 35F25, 35F30, 35Q05, 76C05.

Introduction. The first proof of uniqueness of a plane subsonic flow of a compressible fluid past an obstacle was given by Bers [1]. This proof utilizes an elaborate mathematical apparatus encompassing some of the most advanced tools of modern function theory. A conceptually simpler proof under somewhat weaker hypotheses and a proof of the Joukowsky force formula, due to the authors [3], make essential use of a general existence theorem and hence cannot properly be called elementary. In this note we show that the uniqueness of a compressible flow and the Joukowsky force formula can be obtained directly from a simple geometrical property of the velocity field defined by the flow. Specifically, we base our proof on the facts that the velocity components u, v of the flow define a quasi-conformal(') mapping of the (x, y) plane, and that a quasi-conformal mapping with dilatation ratio not exceeding K = 1/M satisfies at each point a Hilder condition with exponent ,. Until recently we would have considered this proof more difficult than our original one. We are, however, now able to refer to the preceding paper [5 ], in which a simple demonstration is given for the needed lemma. We note, further, that a simpler form of the arguments in [5] suffices for the present paper. The problem of proving uniqueness of a flow past an obstacle P can be divided into two parts; to prove the uniqueness if the velocity at infinity and circulation are both prescribed, and to prove that if P has a corner or cusp T the circulation is uniquely determined by the condition that the speed is finite at T. We shall settle both points here, but for the second we need the additional assumption, not used in [1] or [3], that the velocity components have bounded derivatives up to P. (This assumption can be avoided by a discussion analogous to that in [3, ?6.2].) By the same method we shall also obtain a new proof, simpler than that of [3], for the force formula of gas dynamics. 1. Uniqueness with prescribed circulation. As in [3 ] we define the velocity potential of a flow past P to be a solution k(x, y) of

In this paper it is proved that the perceptron error-correction procedure stays bounded, even when no solution to the system of linear inequalities exists. This supplements earlier papers by B. Efron and by M. Minsky and S. Papert.

This paper adapts the method of F. Riesz to the proof of certain general ergodic theorems for Abelian semi-groups of operators on a Banach space to itself. The main features of the method are that no measurability conditions are imposed on the semi-group under consideration and that consistent use of the second conjugate space and its compactness properties make it possible to replace the compactness conditions often imposed by a more natural restriction on the transforms of points. Theorem 1 and the various supplementary results include as special cases theorems of Lorch [10], Dunford [7], Yosida [15], F. Riesz [12], and Cohen [6]. It overlaps the work of Alaoglu and Birkhoff [3] at those points where they consider Abelian cases; for example, Corollary 8 is a great generalization of their Theorem 5. Section 1 contains some introductory material on conjugate spaces and adjoint operations. Section 2 introduces bounded Abelian semi-groups of operators and near invariance of a system of set functions on such a semi-group; this section also contains the principal theorem (Theorem 1) of the paper. The form of this theorem raises three questions ((A) to (C) at the beginning of ?3). The answer to (A) shows, among other things, that every Abelian semi-group has a property much like "ergodicity" in the sense of Alaoglu and Birkhoff; Theorem 3 is the main result here. The answer to (B) again indicates the importance of reflexivity in theorems of this type; Corollary 8 is one example. Two special cases of (C) give a generalization of Dunford's theorem (Theorem 5) and a theorem on bounded Abelian semi-groups of projections (Theorem 6) which has not, so far as I know, been considered before. 1. Some properties of Banach spaces. If B is a Banach space(2), let B* be the set of all linear-that is, additive and continuous-real-valued functions on B. If, for A in B*, |I|3 =SUpjjbJJ 0, k a positive integer, and I1, . , f3k in B*. With this topology B is a linear topological

Introduction. This paper is mainly concerned with the characterization problem for bounded spectral operators (see [5]). We restrict the discussion to scalar type operators with real spectrum, which we call pseudo-hermitian operators (p. h.). The criteria are given in terms of certain properties of the exponential group generated by the operator. The simplest criterion of this kind is for Hilbert space: a bounded linear operator S is p.h. if and only if || eitS| ? M < oo for all real t. This statement is false for Banach spaces, and even for reflexive Banach spaces. In the latter case, we give additional necessary conditions on the group eitS, which together with its uniform boundedness are both necessary and sufficient for S to be p.h. Another useful criterion is the following: a bounded linear operator S on a reflexive Banach space is p.h. if and only if for everyf in L1(R) (R the real line), we have II fRf(t)e itsdt || _ M || f l,,, where the norm on the left is the operator norm, f is the Fourier transform off and || * is the sup norm. Unlike the general theorems in [5] about the spectrality of an operator, the characterizations obtained in this paper are easily applied to analytically given operators on concrete Banach spaces. An example is discussed in ?4. Applications are discussed at the end of ?5. ?2 deals with an improvement of a theorem of Foguel [7] about the resolutions of the identity of sums and products of commuting spectral operators. ?1 contains preliminaries. The characterization problem for spectral operators is discussed in ??3 and 5.

Let S be a set in a two dimensional Euclidean space E2. Such a set S is said to be arcwise convex [5] if each pair of its points can be joined by a convex arc lying in S. A convex arc is, by definition, an arc which is contained in the boundary of a plane convex set. In two previous papers, [5; 6 ], the author studied certain properties of closed arcwise convex sets. It is the purpose of this paper to study an interesting class of sets which satisfy the three point arcwise convexity property, defined as follows: DEFINITION 1. A set SCE2 is said to have the three point arcwise convexity property if each triple of points x CS, y CS, zCS is contained in a convex arc belonging to S. It should be observed that the above property implies there exists a convex arc in S having two of the three points x, y, z as end points and the third point in its interior. DEFINITION 2. A convex curve, as distinguished from a convex arc, is a closed connected portion of the boundary of a plane convex set. A convex curve may have two, one, or no end points, and it may be bounded or unbounded. The following theorem characterizes the closed sets in E2 which have the three point arcwise convexity property.

Several authors have given definitions of an end point of a continuum, making use of properties which are possessed by an end point of a straight line interval, but which are not possessed by any interior point of the interval. We shall show in the present paper that the properties used in these so-called "definitions" are not logically equivalent, and shall determine the logical relations which do exist among these properties under various conditions. In Part 2, we shall show the equivalence of a number of these properties in the case of a continuous curve. In Part 3 is shown the equivalence with the first set, of two additional properties in the case of a continuous curve of a special type. In Parts 4 and 5, we determine the logical relations existing among certain of these properties in the case of a bounded continuum. Finally, in Part 6, some theorems are proved concerning points having one or more of the given properties. In this paper we shall consider only plane point sets, although in a number of cases it is obvious that our results are true in space of any number of dimensions. In regard to the use of the word "end point," we intend hereafter to use this word only in the sense of a point of a continuous curve satisfying Wilder's definition or one of the other definitions equivalent to it, in other words, a point having any one of the properties 1-7 given in Part 2. We shall not use the word "end point" in referring to a point of a continuum which is not a continuous curve. The examples of Part 4 show that a point of a continuum may have certain of the given properties and yet be so placed with respect to that continuum as hardly to deserve the name of "end point."

on the range space, the class of semigroups which permit the two types of invariant extensions is contained in the class of functions which have invariant means definable on the associated Banach space of bounded, real-valued functions defined on these semigroups. Further it was shown that every semigroup known to have an invariant mean also permitted the two types of extensions. When structure in addition to the minimal required structure is placed on the range space, e.g., when the space is an ordered subspace of the conjugate space of an ordered linear space with reproducing cone, a semi-group which has an invariant mean also has the two extension properties relative to these particular range spaces. Many of the standard function spaces satisfy these conditions. In addition this paper will exhibit conditions which guarantee the continuity of these extensions in the case when linear topological spaces are considered and make some application to real-valued functions. ?11 outlines definitions and results preceding this paper. ?III contains theorems on the existence of invariant extensions for a given semigroup G with an invariant mean and a given range space V. Various extra conditions added to the known necessary condition that V be a boundedly complete vector lattice are sufficient for these extensions. Some converse results are also given under which the existence of some sort of invariant extension implies that G has an invariant mean. ?IV contains examples of spaces satisfying the conditions imposed on the space V. ?V discusses continuity of the extensions when order and topology are both present. ?VI presents applications of the theorems of ??III and V to some of the examples of ?IV.

Introduction. This paper is concerned with extremal problems in the family of bounded analytic functions in a multiply-connected domain D, and it is concerned with extremal problems for the mean modulus f cIf Ids of meromorphic functions f in D taken over the boundary C of D. These two types of problems are shown to be closely related, and solutions are obtained simultaneously for both types by a method of contour integration. An altogether analogous method was exploited by Grunsky in his thesis [8](2) for the investigation of schlicht functions in a multiply-connected domain. Thus it is interesting to remark that the fundamental distortion theorems of schlicht conformal mapping theory have been developed by Grunsky by a method which we are able to apply here to obtain the fundamental distortion theorems for bounded functions. It will appear, then, that the generalization of Schwarz's lemma to multiply-connected domains and the generalization of the Koebe distortion theorem can be carried out by a unified technique(3). We shall find in addition to this that, while the recent papers of Bergman and Schiffer [5, 16] have developed a close relationship between the theory of schlicht canonical mappings and the Bergman kernel function [3], we are able to develop here a relationship between the theory of bounded functions and the Szego kernel function [19]. Thus the Szego kernel function does for the theory of distortion of bounded functions what the Bergman kernel function does for the theory of distortion of schlicht functions. We point out that both these kernel functions are actually differentials, and that in the Szego case one is dealing with length and in the Bergman case one is dealing with area. Thus the mean modulus fc f I ds, or ffD IfI 2dxdy, should be thought of as a length, or area, and not as a mean modulus. All these remarkable relationships are brought to light by using the simple boundary relations satisfied by the classical domain functions, such as

2. Preliminary definitions and theorems. Throughout this paper all integrals considered will be Hellinger [1i type limits of the appropriate sums (the definitions, theorems and proofs of this paper can be extended to "many valued" functions). Thus, if K is a real valued function of subintervals of the number interval [a, b], the existence of f[a,bjK(I) is necessary and sufficient for the existence of f1K(J) for each subinterval I of [a, b]. Furthermore, the function F of subintervals of [a, b], defined by F(I) =fK(J), is additive. If K is a real valued function of subintervals of [a, b], then the statement that K is 1-bounded on [a, b] means that there is a subdivision D of [a, b] such that the set of sums EEK(I), where E is a refinement of D and the sum is taken over all intervals I of E, is bounded. This implies that if I is an interval in a refinement of D, then the set of sums EQ K(J), where Q is a subdivision of I and the sum is taken over all intervals J of Q, has a least upper bound L(I) and a greatest lower bound G(I). We now see that if each of R and T is a refinement of D, and S is a refinement of each of R and T, then ER G(I)-< Es G(I) -< s K(I) ? Es L(I) ? E2T L(I). From this it follows that each of f(a,b]G(I) and f[a,b]L(I) exists, that f[a,b]G(I) < f[a,b]L(I), and that f(a,b]K(I) exists if and only if f[a,b]G(J) f[a,b]L(I). We state without proof a theorem of Kolmogoroff [2].

In a recent paper, Felix E. Browder discussed continuous self-mappings on a metric space, satisfying a func- tional inequality. Browder gave sufficient conditions such that the successive approximations of any point for such mappings con- verge to a unique fixed point. In the present paper, Browder's result is extended to a commutative semigroup of mappings and also to single mappings that are not necessarily continuous and satisfy a weaker form of the functional inequality. 1. It is the purpose of this paper to generalize the following result of Felix Browder (2), under the assumption that the set M there is closed. Theorem 1. Let (X, d) be a complete metric space, M a bounded subset of X, Ta mapping of M into M. Suppose there exists a monotone nondecreasing function \p(r) for r = 0 with \f/ continuous on the right, such that if/(r) 0, while for x, y

The two main results in this paper are analogues of the Stone-Weierstrass theorem for real-valued functions, obtained by using different function space topologies. The first (Theorem 2.3) is a Stone-Weierstrass theorem for unbounded functions. The second (Theorem 3.6) is a theorem for bounded functions; it is stronger than the usual theorem because the topology is larger than the uniform topology. In fact it is the strongest possible theorem in a sense made precise in Theorem 3.5. However, the usefulness of the result is, in both cases, limited by the fact that the topologies used are far from being as well behaved as the uniform topology. Some specific defects of these topologies are discussed in Proposition 2.4, Proposition 3.9, and Example 3.10. (The topologies are defined in ?1.) The formulation of the Stone-Weierstrass theorem on which the analogues in the present paper are based is the following: If X is a completely regular topological space, then X is compact iff every strongly separating subalgebra of C*(X) is u-dense, where u denotes the uniform topology. In [5] Lorch has studied a more general class of function algebras than those of the form C*(X). All of the results for bounded functions in this paper are formulated in this more general setting (see ?1).

The present article is intended as a survey of the title subject. The material is closely intertwined with all branches of harmonic analysis. Historically, the rigorous foundations of the theory arise in Riemann's treatment of trigonometric series, and spectral theory is essentially equivalent to the study of formal multiplication. The original motivation for the modern treatment, due to Wiener, Carleman, and Beurling, came from the study of integral equations with convolution kernels. There are applications to linear partial differential equations with constant coefficients, and there is a very close connection with problems about entire functions bounded on a line. My own concern with the topic commenced with questions about the theory of approximation for multiple Fourier transforms. When I first looked at the subject the basic material did not appear very well organized, and certain elementary facts were not recognized as immediately obvious to experts in the field. Thus, in 1954, I set about to record what was known with certain useful additions. Shortly after the original version of this work was finished in 1956, the paper of DOMAR appeared. (An author's name in small capitals indicates a reference to the bibliography.) There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling. I have revised the paper in an attempt to suppress details which may be found elsewhere. Also, I have taken advantage of more recent work by others and myself to improve the material of the last half of the paper. After a preliminary section introducing much of the notation and basic definitiong, the contents of this paper are divided into six parts. The headings are: 1. The spectrum. 2. The point spectrum. 3. Potential theory and spectral analysis. 4. The spectral synthesis problem. 5. Representations. 6. Examples of spectral synthesis.

In Part I of this paper we construct a bounded-truth-table-complete set which is not creative, and in Part II we construct an infinite and coinfinite recursively enumerable (r.e.) splinter which is not a cylinder. Although these results appear unrelated, the proofs given here are similar and so have been included in a single paper. The two parts may be read independently of one another. Each of these results yields as corollary the existence of pseudo-creative sets which are not cylinders. In fact, in Part II we construct a pseudo-creative set S for which there is no total recursive function f such that x C S implies f (x) C S-{ x } and x C S' implies f (x) C S'-{ x } .

The Chaplygin equation, which arises in the study of transonic flow, is one of the simplest equations of mixed type. It was shown that with appropriate hypotheses on the function h a problem of this type could be stabilized (i.e., the solution made to depend continuously on the data) by restricting the solution to lie in the class of functions whose L2 integrals are bounded by some prescribed constant M. Similar restrictions have previously been shown (see e.g., [2], [7]) to guarantee continuous dependence in certain improperly posed problems for elliptic and parabolic equations. However, little study has up to now been devoted to improperly posed problems for equations of mixed type. For that matter one is rarely able to determine just what is a well set problem for such an equation. In this paper we demonstrate that the convexity arguments which led to stability inequalities for the Chaplygin equation [6] actually can be carried over with some modification to yield stability inequalities and error bounds in improperly posed problems for a wide class of quasilinear partial differential equations of mixed type. We shall be concerned here only with the question of obtaining such inequalities. We leave aside the difficult question of existence. The particular equation which we consider is by no means the most general one amenable to the convexity method. We restrict our attention to a relatively simple equation but one which exhibits most of the troublesome features which would appear in the more general operators. A survey of the literature on improperly posed problems will appear in a forthcoming paper by Payne [5]. In this paper we consider the following initial-boundary value problem for a cylindrical domain Q = D x (0, Y), where D is an arbitrary n-dimensional domain with smooth boundary D

Part I of this paper has to do with connected sets of cut pointst of a given continuous curve. It is shown that, in the plane, any two points belonging to a connected set K of cut points of a given continuous curve M lie together in an arc which is common to K and a point set consisting of the boundaries of a finite number of complementary domains of M. G. T. Whyburnt calls attention to the fact that from his results it follows that K is arcwise connected. To show that any two points of K can be joined by an arc which is common to K and a point set consisting of the boundaries of a finite number of complementary domains of M, is the object of Part I. Part II has to do with a totally disconnected closed subset K of a given plane continuous curve M no subset of which disconnects M. R. L. Moore? has shown that, in the plane, any two points not belonging to a bounded continuous curve can be joined by an arc which does not disconnect the continuous curve. The object of Part II of this paper is to show that if, in the plane, M is a bounded continuous curve which contains no domain, and K is a closed and totally disconnected subset of Al, such that no subset of K disconnects M, then there exists an acyclic continuous curve T, containing K, such that (1) all the end pointsll of T belong to K and (2) the point set M T is totally disconnected and M-T is connected. I wish to acknowledge my indebtedness to Professor R. L. Moore, not only for the suggestions of the problems treated, but also for his many helpful crltici;crn ;P their solutions. To him is due, in a large measure, the success

THnORUM I. If the points z1, Z2, Z3 vary independently and have circular regions as their respective loci, then the locus of the point Z4 defined by the real constant cross ratio X=(Zl, Z2, Z3, Z4) is also a circular region. It is the purpose of the present paper to consider generalizations of and other results related to Theorem I, primarily the determination of the locus of the point Z4 defined as in Theorem I, when zi, Z2, Z3 vary independently so as to have certain prescribed loci. Thus one may raise the following question: If two variable points lie respectively on two fixed circles, where does the mid-point of their segment lie? The answer is contained in Theorem VI. Or again, if the loci of the points zi, Z2, Z3 are regions each bounded by a number of circles, what can be said of the locus of z4? The answer is given by Theorem VII. All the results proved concerning such loci as these can be interpreted in terms of the roots of the jacobian of two binary forms. Such an interpretation is given in ?12. The chief method used in the present paper to determine the locus of Z4 in any given case is that suggested in I (pp. 102, 103, footnote), namely the determination of the locus of z4 when z1 and Z2 are held fast and Z3 varies over its locus; the determination of the locus of this locus of z4 when z1 is kept fixed but Z2 varies over its locus; and finally the determination of the locus of this new locus of Z4 when z1 varies over its prescribed locus. In the present paper this method is carried through geometrically. The same method has been used by Professor A. B. Coblet to determine analytically the locus of Z4 in Theorem I

In this paper we shall confine ourselves to a two-dimensional euclidean space which we shall denote by the symbol S. R. L. Mooret has given the following extension of a theorem by Zoretti:t If K is a bounded maximal connected subset of a closed point set M and does not separate space, and e is a positive number, then there exists a simple closed curve containing no point of M, such that the interior of this curve contains K but contains no point whose distance from K is greater than e. Zoretti's conclusion is weaker than Moore's in that it says nothing about the points on the interior of the curve surrounding K. On the other hand, Moore finds it necessary to use a stronger hypothesis than Zoretti's; namely that K does not separate the plane. In this paper we consider among other things the question of the conditions involved in a combination of Zoretti's hypothesis and Moore's conclusion, and show in Theorem I that a result analogous to

for almost all t. In this definition it is assumed, of course, that the integral on the right-hand side of (1.2) is in the domain of A. In a recent paper [6], Friedman and Shinbrot have studied the equation (1.1) even in the more general case where xo and A depend on t and -, respectively. They proved theorems of existence, uniqueness, differentiability and asymptotic behavior of solutions. They also constructed fundamental solutions and derived asymptotic bounds for them. We recall [6] that if xo is in the domain of A4, for some ,u>0, then the solution of (1.1) is given by S(t)xo. The purpose of the present paper is to derive monotonicity theorems for solutions of (1.1). We shall generalize some of the monotonicity theorems of Friedman [2] (see also [4]) from the case X= R1 (R1 the one-dimensional Euclidean space) to the case where X is any Banach space. In ?2 we give some auxiliary results. These results are concerned with Volterra equations in one-dimension (i.e., X=R1). In particular, we study the behavior of the solutions with respect to a certain parameter. In ?3 we give an integral formula for S(t) in case A is a bounded operator. For A unbounded, we construct a fundamental solution as a limit of fundamental solutions corresponding to the bounded operators A(I+ A/n) -1. We prove that the fundamental solution coincides with the fundamental solution of [6, Chapter 1]

Introduction. In Cameron's paper [1] there appears a "rectangle formula" (there denoted by Theorem 1) by means of which the Wiener integral of F[x(.)], defined on the space of continuous functions on [0, 1 ] such that x(O) = 0, can be approximated by means of an n-fold Riemann integral provided F is sufficiently smooth and dominated by a suitable integrable functional. The formula employs a particular C.O.N. set of functions {ai(s) } 1 (the odd harmonic cosine functions.) A generalization of this formula appears in Cameron's unpublished notes and appears with proof in the author's [2], (there denoted by Theorem 2). The generalization allows for the use of an arbitrary C.O.N. set {ai(s) } of B.V. but suffers the defect that it is not known to hold except for bounded functionals. It is the purpose of this paper to show that if the {ai(s) }'s of this generalization are restricted to certain Sturm-Liouville sets of functions (among which sets is included the odd harmonic cosine functions), then the restricted generalization is applicable to suitably dominated (not necessarily bounded) functionals. Let w(s) >0 have continuous fourth derivative on [0, 1] and let {As} and {ai(s) } be the sets of characteristic numbers and normalized functions of the Sturm-Liouville problem: { [l/w(s) ]x'(s) }' +-Xa(s)=0, a'(O)=ca(1)=0 on [0, 1]. Let C be the space of continuous functions x(s) on [0, 1] such that x(O) = 0, let fli(s) =f'ai(t)dt and for x ()EC let ci = foea (s)dx (s): i = 1, 2, 3, * * (c, is of course a functional of x(.)). The main result of the paper is the following.

1.1. Inherent in the definitions of one of the basic concepts in modern theory of the area of a continuous surface (see bibliography) there is a question which this note intends to resolve. The situation will be described presently. Since the question first comes to light in the first two chapters of the paper by Rado and Reichelderfer titled A theory of absolutely continuous transformations in the plane-hereinafter cited as R2 [1] (1)-the reader is requested to consult that paper freely while reading this note. Definitions, theorems, and-as far as practicable-notations in R2 [1 ] will be used in the sequel. 1.2. Let S denote a bounded point set in the uv-plane. Given a pair of finite-valued, real-valued functions x(u, v), y(u, v), defined, bounded, and continuous on S, these determine a bounded continuous transformation T:x=x(u, v), y=y(u, v), (u, v)ES, from the set S into the xy-plane (cf. R2 [1, 1.8 ]). For conciseness, complex notation z = x+iy, w = u+iv is introduced, and T is written in the form T:z=t(w), wCS, where t(w) =x(u, v)+iy(u, v), u+iv=wES, is finite-valued, complex-valued, bounded, continuous on S. If E be any set in the w-plane, the transformation T associates with each point g in the z-plane a crude multiplicity with respect to E, N(a, T, E), defined to be the number (possibly + co) of distinct points o in the set E T-F(g), where T-1(g) denotes the set of all points o in S for which t(W) = (cf. R2 [1, 1.10, 1.8]). 1.3. Using the crude multiplicity, Banach [1] and Schauder [1] have developed a beautiful, though restrictive, theory of continuous transformations, which has been completed in many respects by Rado [2 ]. But the crude multiplicity has been found to be generally too large to play an effective role in modern theory of the area. A study of the works of Ge6cze led Rado to the idea of replacing the crude multiplicity by a generally smaller multiplicity (see Rado [1]), which has come to be known as the essential multiplicity. 1.4. The intuitive idea of the essential multiplicity is embodied in the following statement. DEFINITION. Let T:z=t(w), wES, be a bounded continuous transforma

In this paper is presented a theory of unbounded operators which admit spectral distributions. The compactness of the support in the bounded case has been replaced by certain growth conditions at infinity. Properties of these operators are studied, including the existence of a Spectral Mapping Theorem. Inttoduction. In [2], Foias introduced the class of bounded operators in Banach spaces which in lieu of spectral measures admit spectral distributions. He proved certain basic properties of these "generalized scalars", such as the Spectral Mapping Thorem. Further investigations of this class of operators and of spectral distributions were made by various authors in [1], [2], [4], [5], [6],1 [8], and [10]. In [5], the present author studied the class of generalized scalars with real spectrum, which were called generalized pseudo-hermitian (g.p.h) operators. These operators were characterized by the elementary condition that the real exponential group generated by such an operator have polynomial growth in the uniform operator norm.2 In attempting to study the spectral properties of the infinitesimal generators of CO-groups with polynomial growth, the author has been led to consider unbounded generalized scalar operators. In this paper is presented a theory of unbounded spectral distributions and of the unbounded generalized scalar operators which they represent. The compactness of the support in the bounded case has been replaced by certain growth conditions at infinity sufficient to insure that the definitions are correct and the resulting class of operators well behaved, in that they generalize both the bounded generalized scalars and the unbounded normal operators in Hilbert space. Some properties of the class are discussed, culminating in a Spectral Mapping Theorem. 1. Basic definitions. DEFINITION 1. Let T be a mapping from the space -9(R2) of complex test functions into the space Y(X) of linear continuous operators on a Received by the editors April 6, 1971. 4MS 1970 subject classifications. Primary 47A65, 47A60, 47B40; Secondary 46F99. I Also considers unbounded operators. 2 Cf. Theorem 2 of [101. (? American Mathematical Society 1972