Let (Ω, α, μ) be a σ-finite measure space. By Lp(Ω, α, μ) or Lp for short we denote the usual Banach space of pth power μ-integrable functions on Ω if 1≤p<+ ∞ and μ-essentially bounded functions on Ω, if p= +∞. In section (2) we characterize conditional expectation, by a method different than those used previously. Modulus of a given contraction is discussed in section (3). If the given contraction has a fixed point, then its modulus has a simple form (theorem 3.2).

Several years ago, C. Long wrote two papers ([3], [4]) that related to F(n) the number of ordered factorizations of n. The second of these papers [4] was devoted entirely to a discussion of conjectured formula for F(n).

Let X and Y be isomorphic normed linear spaces and let i ℒ denote the set of isomorphisms from X to Y (i.e. bounded linear mappings from X onto Y which have bounded inverses).

Recently, Dunham has given examples for 1-parameter and 2-parameter Chebyshev sets which are not suns. In this note 2-parameter sets with these properties are described.

When studying the old problem whether Chebyshev sets are always convex, Klee [10] introduced certain sets which were called suns by Efimov and Stechkin [7]. Recently, in two shorts notes Dunham [4, 5] has given examples of 1-parameter- and 2-parameter-sets which are Chebyshev sets but not suns (cf. also [3]). The examples refer to Chebyshev sets in containing an isolated point.

In the definition of the Pn-integral [2] there is a difficulty with the condition Bn-2 ([2], p. 150) since it is not linear on the set of major and minor functions. As a result, the proof of Lemma 5.1 [2] fails since the difference Q(x)—q(x) need not satisfy the conditions of Theorem 4.2, [2].

Limit vector spaces ('limitierte Vektorräume') were defined by Fischer [1] and concepts such as continuity, compactness, etc. were introduced and studied by him and others, e.g. by Kent [3]. In this paper the concepts of precompactness and boundedness in limit vector spaces are studied. It is shown that most of their properties in topological vector spaces hold in limit vector spaces.

Il est bien connu que l'ensemble des fonctions de Baire de ℝ dans ℝ est égal à l'ensemble des fonctions boréliennes de ℝ dans ℝ. Dans cet article, nous généraliserons ce théorème de deux façons: premièrement, en étudiant plutöt des fonctions de X à Y, deux espaces topoiogiques normaux; et deuxièmement, en utilisant les suites généralisées de cardinalité donnée.

In previous papers (see [1, 2, 3, 4]), we solved the following functional equation:

1

wheref=f(z) is an entire function of a complex variable z and x, y are complex variables.

The passage from ordinary to partial differential equations is a difficult one. While many properties of ordinary differential equations admit analogies in the framework of partial differential equations, the techniques used to establish them may bear little resemblance to those which suffice in the simpler setting.

Dans cet article, nous nous intéressons à trouver des conditions suffisantes au problème d'approximation polynômiale généralisée dans certains espaces seminormes. Notre résultat principal est basé sur un théorème d'unicité de Malliavin et sur la théorie de la représentation de Choquet.

A Lidstone series provides a (formal) two-point expansion of a given function f(x) in terms of its derivatives of even order at the nodes 0 and 1 and takes the form.

Soient R un anneau commutatif unitaire, S une R-algèbre associative unitaire, A et B des sous-R-modules de S. Nous étudions des situations dans lesquelles HomR(A, B) peut être emboîté dans S. Les définitions et le résultat de base se trouvent dans la section 1; dans la section 2, nous examinons des applications à l'étude des structures d'algèbre que l'on peut définir sur A; la section 3 contient des exemples.

For a simple Lie algebra L over the complex numbers ℂ all irreducible representations admitting a highest weight have been constructed and characterized for example in [3, 6]. In [1] Bouwer considered the family of all irreducible representations of L admitting at least one one-dimensional weight space (this includes, of course, all those having a highest weight space) and showed, by construction, that this is a strictly larger class of representations.

In measure theory and probability it is often useful to be able to extend a set function g to a measure μ. One situation in which such an extension arises is that of obtaining limit measures for inverse (or projective) systems of measure spaces ([1], [5]).

Let {Xi:i=1, 2,…} be a real strictly stationary process (defined on a probability space (Ω, A, P)) which has absolutely continuous finite dimensional distributions (with respect to Lebesgue measure) and satisfies the ϕ-mixing condition: Let and denote the sub-cr-fields generated, respectively, by {Xi:i≤k} and {Xi:i≥k+n}; then, for each k≥1 and n≥l, and together imply.

Let R and S be rings with 1, G a group and RG and SG the corresponding group rings. In this paper, we study the problem of when RG≃SG implies R≃S. This problem was previously investigated in [8] for the case where G is assumed to be infinite cyclic. The corresponding question for polynomial rings, namely, when does R[x]≃S[x] imply R≃S, has been considered by several authors, particularly Coleman and Enochs [3].

We are concerned with the nth order differential equation y(n) = (x, y, y′, …,y(n-1)), where it is assumed throughout that f is continuous on [α,β) × Rn, α < β≤∞, and that solutions of initial value problems are unique and exist on [α, β). The definition of the first conjugate point function η1(t) for linear homogeneous equations is extended to this nonlinear case. Our main concern is what properties of this conjugacy function are valid in the nonlinear case.

For each Hausdorff space X, let FX be an Hausdorff extension of X. The existence of the largest subcategory of HAUS on which F s a functor and an epi-reflection is investigated.

For α>0 the Riemann-Liouville Integral J(α) is given for suitable functions g by

1

For a variety of function spaces (e.g., C[0, 1] or Lp(0, 1) with p≥1) this defines a C0 semigroup which has been extensively studied (cf., e.g., [3]).

In this note, we prove analogues of the classical maximum modulus theorem and Schwarz lemma, for sequence spaces. We begin by stating these two results in a convenient way; that is for the unit disk and functions of bound one.