Space Of Linear Functionals Research Articles

On Some Applications of Graph Theory III

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].

Some metric and topological properties of certain linear spaces of measurable functions

Some metric and topological properties of certain linear spaces of measurable functions

A note on the inheritance of properties of locally convex spaces by subspaces of countable codimension

A locally convex space E is said to be c-barrelled if every countable weak* bounded subset of its topological dual E' is equicontinuous; to have property (C) if every weak* bounded stubset of E' is relatively weak* compact; to have property (S) if E' is weak* sequentially complete. If a locally convex space possesses any of the above properties, then so do all of its linear subspaces of countable codimension. Examples are furnished to show that the mentioned properties are distinct from each other. In [4], the authors proved that a linear subspace of countable codimension in a barrelled space is barrelled. In this paper, we see that the properties of a space being w-barrelled and having weak*sequentially complete dual are inherited by subspaces of countable codimension. Only a partial result is obtained concerning the seemingly most important property, that of being a Mackey space. The paper concludes with examples to demonstrate that the various concepts discussed are, in fact, disjoint. 1. The notation will be essentially that used by J. Horvath [3]. If (E, F) is a dual pairing (E and F not necessarily separating points), then o-(E, F) will denote the topology on E of pointwise convergence on F. The polar A' of a subset A of E is the set {If F: (a,f)l ? 1 for all aEA }. -r(F, E) will denote the topology on F of uniform convergence on o-(E, F)-compact subsets of E. The vector space of continuous linear functionals on a locally convex space E will be designated E'. A (not necessarily Hausdorff) locally convex space E is said to be barrelled if every closed, balanced, convex, absorbing subset of E is a neighborhood of 0 or, equivalently, if every o-(E', E)-bounded subset of E' is equicontinuous. A locally convex space E is said to be w-barrelled if every countable, o-(E', E)-bounded subset of E' is equicontinuous; to have property (C) if every o(E', E)-bounded subset of E' is relatively o-(E', E)-countably compact; to have property (S) if E' is o-(E', E)-sequentially complete; to be a Mackey space if it has the topology r(E, E'). The codimension of a linear subspace M of Presented to the Society, January 24, 1969; received by the editors June 10, 1970. AMAS 1970 subject classifications. Primary 46A05, 46A15.

An application of distribution theory to Martin's method in potential scattering

AbstractIn order to study the convergence of a series arising from the use of Martin's method in potential scattering a framework consisting of a modifidation to Schwartz's formulation of distribution theory is set up. This modification makes essential use of norms on the spaces of test functions and linear functionals. By making the norm on the space of test functions satisfy extra axioms a bound is obtained on the norm of a generalized convolution of two linear functionals, and this bound is used to prove the convergence of the series. The proof of convergence is generalized to cover the multi-channel problem. The method used may also be applied to many other physical problems.

Zero-One Laws for Gaussian Processes

Some zero-one laws are proved for Gaussian processes defined on linear spaces of functions. They are generalizations of a result for Wiener measure due to R. H. Cameron and R. E. Graves. The proofs exploit an interesting relationship between a Gaussian process and its reproducing kernel Hilbert space. Applications are discussed.

Zero-one laws for Gaussian processes

Some zero-one laws are proved for Gaussian processes defined on linear spaces of functions. They are generalizations of a result for Wiener measure due to R. H. Cameron and R. E. Graves. The proofs exploit an interesting relationship between a Gaussian process and its reproducing kernel Hilbert space. Applications are discussed.

Duality of linear spaces of functions and nuclearity of solution spaces

Duality of linear spaces of functions and nuclearity of solution spaces

HARMONIC ANALYSIS OF FUNCTIONS ON SEMISIMPLE LIE GROUPS. II

A theory of harmonic analysis is developed for the class of functions (fundamental and generalized) with compact support on an arbitrary semisimple complex connected Lie group. Duality theorems are proved for the linear topological spaces of finite functions most often encountered in analysis (infinitely differentiable finite functions, finite functions in L² , and finite generalized functions). All results are analogs of the standard theorems of Paley-Wiener type in harmonic analysis on the line.

Non-local variational mechanics—IV stationarization with several dependent variables

THIS paper is a direct continuation of the first three paperst in this series. The notation of the first three papers is assumed throughout and is used without further note. Citation of results from previous papers is made by equation number. The numbering of this paper follows that of the first three papers in sequence. IO. STATlONARIZATlON AND THE NON-LOCAL EULER-LAGRANGE OPERATOR The types of problems we now wish to study are those involving two or more dependent functions. In the case of only one dependent function, we found that it was sufficient to consider the linear spaces of functions of class C or of class Cl with the uniform convergence norm. We now consider the situation wherein we must use vector spaces of function spaces. Let {@*(xk)}, A = 1,. . ., N, denote an order collection of N functions of class C or of class Cl on D*. Such collections of functions form a linear space with the usual definitions of addition and scalar multiplication. If each of the {@*(I?)} is of class C on D * we define the norm by

Weighted polynomial approximation on the integers

We prove here some polynomial approximation theorems, somewhat related to the Szasz-Mfintz theorem, but where the domain of approximation is the integers, by dualizing a gap theorem of C. l ~ Y I for periodic entire functions. In another Paper [7], we shall prove, by similar means, a completeness theorem ibr some special sets of entire functions. I t is well known (see, for example [l]) tha t i f E is the space of all entire functions in the topology of uniform convergence on compact sets, then the dual space of continuous complex-valued linear functionals on E may be represented as E0, the space of entire functions of exponential type. Now let E (1) b e the space of entire functions of period 1. Then it may be shown that the dual of E (1) can be represented as E0 (l), where E0(1) is the following quotient space of E0: define / ~ g for functions 1, g e E0 i f / _ g is a multiple of sin z~z, and let E0 (l) be the space of equivalence classes of E0 modulo this relation of equivalence. Now E0(1) is apparent ly the same space as the space of restrictions of functions in E0 to the integers Z. Each such restriction is just a two-sided sequence of complex numbers, of at most exponential growth. ConVersely, it is easy to interpolate any such sequence by an entire function of exponential type. Thus, the dual of E(1) is just the space of all such sequences. Actually, we establish this identification by another procedure. To any theorem about periodic entire functions will correspond a theorem about the space of sequences described above. In [6], C. RI~NYI proved an interesting gap theorem, reproduced below. We show by means of duality tha t certain theorems of polynomial approximation are equivalent to this theorem. The domain of approximation is the integers in one case, and the positive integers in another ease. To our knowledge, the problem of polynomial approximation on the integers has not been Considered except in the note [3]. We know of no direct proof of our results. Ultimately, the R ~ Y I result depends on a simple application of Rolle's theorem. I t Would be of interest to have more precise gap theorems than the R~NYr theorem and also to have direct proofs of the results we prove by means of it.

Singular Behavior of Zonal Hyperspherical Harmonic Series

An integral operator, defined in a bounded domain, is constructed to generate a complete set of particular solutions, namely, the hyperspherical harmonics, for the Laplace equation in four variables from the linear space of analytic functions of two complex variables. An inverse operator is also constructed. The singular behavior of the solutions is investigated when the corresponding properties of the analytic functions are known.

Point derivations in certain sup-norm algebras

1. Let A be a closed point-separating subalgebra of C(X) containing the constants, where X is a compact Hausdorff space. MA will denote the space of multiplicative linear functionals p on A, and to each such p we associate its kernel A,. The A, are precisely the maximal ideals of A. Under certain hypotheses, it is known that analytic discs can be embedded in MA. Wermer [W1] showed that if A is a Dirichlet algebra on X, then each Gleason part of A is either a single point or an analytic disc. Hoffman [H] then generalized Wermer's result to logmodular algebras. Finally Lumer [L] observed that the conclusion is really local: if p has a unique representing measure on X, then the part for A containing p consists either of p alone or of an analytic disc. Our objective in this paper is to take the weakest of these possible hypotheses, namely Lumer's, and show that in a broader sense the analytic disc at p, if there is one, really does account for all the analytic structure at (p. Specifically, we show that all the bounded derivations and higher derivatives of A at cp are just differentiations with respect to the analytic structure of the analytic disc.

Properties of the sequence of closed powers of a maximal ideal in a sup-norm algebra

1. A sup-norm algebra is a closed point-separating subalgebra A of C(X) containing the constants, where X is a compact Hausdorff space. MA will denote the (compact Hausdorff) space of multiplicative linear functionals p on A in the Gelfand topology, and to each such p we associate its kernel A,; the A, are precisely the maximal ideals of A. X can be viewed as a closed subset of MA, and there is an isomorphism f -> of A into C(MA) such that f' X=f. The isomorphism is given by I(p) =(pf). We will frequently write f for f! SA will denote the Silov boundary of A, that is, the smallest closed subset of X on which each function in A attains its maximum modulus. Eachfs A has the same supremum over SA, X, and MA; this common supremum is the norm off, written 11f 1. We intend to study the ideals (A')-. [Note: If I is an ideal in A, In is the ideal generated by products fi... fn with f, E I if n > 1, and I? = A; also I denotes the closure of L] Evidently (An) D (A n+ 1)and if the inclusion is an equality for some n, then it is an equality for all subsequent n. A problem of the sort that will interest us-indeed, in great part the stimulus for this research-is to determine whether one can preassign the cutoff point in this chain of inclusions: given n, is it possible to have (A n) =A (A n + 1) -but (An + 1) -= (A n + 2) -? More generally, how much control do we have over the sequence of numbers {dim ((An) /(An + 1)-): n= 1 2,.. . }? While we will not give an explicit arithmetic answer, we will reduce the problem to a purely algebraic one. In particular, it is possible to obtain any nonincreasing sequence of nonnegative integers, and hence to preassign the cutoff point. We now will begin to formulate the problem we will actually solve. First, consider a graded algebra (all algebras in this paper are commutative algebras over the complex field C) Q = ? (1) Qn and suppose that Qo consists of scalar multiples of an identity for Q, and that Qo and Q, generate Q. If {F,: y E L'} is a basis for Q, and R = C[{XY : y E ]}] is the polynomial ring in P variables with its usual graded algebra structure, the linear extension of the map X Y n Yn * . is a graded algebra homomorphism of R onto Q. Thus Q is isomorphic as a

Abstract Köthe spaces. III

In this paper I shall be considering the space of linear maps between two (perfect) Riesz spaces, and shall show how certain topological properties of these maps are related to the natural order structure of the space. The fundamental result is (e) of section 4, certain special cases of which have been treated in (4) and (5), using a less elliptic method of proof. Probably the most interesting new result in the present paper is section 10 in the special case of both L and M× being L1 spaces (so that |σ| (L, L×) and |σ| (M×, M) are the norm topologies ((2 b), section 7) and Λ(L; M×) is the space of norm-continuous linear maps).

Level sets and continuity of conjugate convex functions

A finite-valued convex function on a nonempty convex set C in F can always be extended to a proper convex function on F by assigning it the value + 0o outside of C. Let F and G be real vector spaces in duality with respect to a bilinear functional (x, y) for x e F and y E G (see [1, p. 48]). We shall henceforth assume F and G have each been supplied with a topology compatible with this duality [1, p. 67], so that each can be identified with the space of continuous linear functionals on the other. Unless explicit notice is given, all questions of closure, continuity and boundedness refer to these given topologies. The formulas

A Property of Algebraic Homogeneity for Linear Function-Spaces

A Property of Algebraic Homogeneity for Linear Function-Spaces

Factorization of polynomials over Banach algebras

Introduction. A Banach algebra in this paper will be understood to mean a commutative, semi-simple Banach algebra with multiplicative unit e. By the carrier space DA of the Banach algebra A, we mean the space of multiplicative linear functionals on A to C, the complex numbers, and it is to be endowed with the usual weak* topology (cf. [6]). For a E A, a denotes the Gelfand transform of a defined on (DA and A will denote the collection of such functions. We set the following notation. x will be used to denote an indeterminate over A as well as over A and C. If x(x) = 7=Oa,ix' is a polynomial over A, let &(x) and ;,(x) denote, respectively, ,i oQX,(2) = 0}, ac(x) A[x], plays an important role in the present paper. Z(oc(x),A) is topologized with the relative product topology from (A x C. The mapping n is defined by r(h, 2) = h, (h, 2) e Z(oc(x), A). The multiplicity function M of a(x) is defined as follows: for (h, 2) E Z(x(x), A), M(h, A) is equal to the multiplicity of A as a root of cth(x) = 0. In ?1 we introduce the concept of M-neighborhood of a point in Z(a(x),A). We say that W c Z(cx(x), A) is a M-neighborhood of (ho, 2A) E Z(oc(x), A) if W is a neighborhood in Z(oc(x),A) of (ho,AO) and if, for each h e p(W), M(ho, 2O) is equal to the sum of the values of M at the points (h,2) in 7K-1(h) nl W. Proposition 1.1 states that M-neighborhoods exist and that they form a base for the neighborhood system at each point of Z(ox(x),A). The remainder of this section contains most of the topological lemmas needed for our work on factorization. Particular attention is paid to the case where Z(o(x),A) contains a compact open subset K (7t(K) = 'DA) on which M is constant. When this condition obtains, K and 4FD decompose topologically and this decomposition in turn forces a(x) to factor. The main factorization theorem (2.1) says that if a(x)eA[x] and if K (nr(K)-=(A) is a compact open subset of Z(a(x), A), then there exists a monic polynomial P(x) e A[x] such that ,B(x) is a factor of o(x), Z(fl(x), A) = K and Z(ax(x)/fl(x), A) = Z(a(x), A) K. A more detailed description of the factorization of monic polynomials follows. For example, it is shown that if A is indecomposable

Lipschitz classes of functions and distributions in $E_n$

The results summarized here are the principle results of the aut h o r s doctoral dissertation presented at the University of Chicago and written under the direction of E. M. Stein. These results will appear soon with proofs. We consider properties of classes of functions and distributions which are characterized by various smoothness and differentiability conditions. Similar classes have been studied by many investigators in recent years. Those papers touching most closely on the results given here are cited throughout this announcement. Special attention, however, is directed to the thorough list of references given recently by Nikolskii in [9]. Our methods are analogous to those of Hardy and Littlewood in their study of analytic and harmonic functions in the unit disc (see [o]) extended to the w-dimensional nonperiodic case by consideration of Poisson integrals of tempered distributions in the same spirit as Stein and Weiss [ l l ] , and Stein [lO]. We shall consistently denote by x and h elements of Eni and by y elements of the positive real axis. By Lp(En) = Lp we mean the normed linear space of measurable functions f(x) for which the norm

On functions satisfying certain differential inequalities

then f(x) = ce-x. In ?2 a new proof is given using Liouville's theorem. So far as I am aware, the method of proof does not apply to any essentially more general problems, depending as it does on unique properties of the analytic function ez. However, I have obtained a few similar results by other methods, and to establish a sufficiently broad framework to contain all cases, it is convenient to pose a somewhat indefinite general problem: PROBLEM A. Let C1 be a linear space of functions; let T be an operator on Ci to C1; and let .| be a norm defined on C1. When does fEC1 and II TkfII _1, k=O, 1, 2, ,imply that fC C2 where C2 is a finite-dimensional subspace of C1? A narrower framework will also be considered: PROBLEM B. Let T be a differential operator and let g be a function such that Tg -g (or +g). When does I Tkf| <gi, k=O, 1, 2, imply f = cg? A very simple result of type A is obtained in ?3 by taking T to be a self-adjoint operator on a Hilbert space. Differential operators are natural choices for T, and some special cases are spelled out using Fourier and Hermite expansions. A corollary of type B is given. In ?4, I consider the possibility of generalizing results for particular operators T to results for conjugate operators of the form STS-1. For Problem B a natural choice for S is an operator which acts in effect to change the variables of the differential equation Tf = -f. By employing this device, results are obtained for example for T=pl(x)D +po(x) and for T=AD2+BD+C using respectively Tagamlitzki's theorem and a Fourier expansion.

Numerical integration of ordinary differential equations based on trigonometric polynomials

There are many numerical methods available for the step-by-step integration of ordinary differential equations. Only few of them, however, take advantage of special properties of the solution that may be known in advance. Examples of such methods are those developed by BROCK and MURRAY [9], and by DENNIS Eg], for exponential type solutions, and a method by URABE and MISE [b~ designed for solutions in whose Taylor expansion the most significant terms are of relatively high order. The present paper is concerned with the case of periodic or oscillatory solutions where the frequency, or some suitable substitute, can be estimated in advance. Our methods will integrate exactly appropriate trigonometric polynomials of given order, just as classical methods integrate exactly algebraic polynomials of given degree. The resulting methods depend on a parameter, v=h~o, where h is the step length and ~o the frequency in question, and they reduce to classical methods if v-~0. Our results have also obvious applications to numerical quadrature. They will, however, not be considered in this paper. 1. Linear functionals of algebraic and trigonometric order In this section [a, b~ is a finite closed interval and C ~ [a, b~ (s > 0) denotes the linear space of functions x(t) having s continuous derivatives in Fa, b~. We assume C s [a, b~ normed by s (t.tt IIxll = )2 m~x Ix~ (ttt. a=0 a~t~b