scholarly journals Decompositions of vector measures in Riesz spaces and Banach lattices

1986 ◽  
Vol 29 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Klaus D. Schmidt
Keyword(s):  
Bounded Variation ◽  
Banach Lattice ◽  
Decomposition Theorem ◽  
Riesz Space ◽  
Jordan Decomposition ◽  
Countable Additivity ◽  
Vector Measures ◽  
Central Result ◽  
Decomposition Theorems ◽  
Natural Way

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.

scholarly journals Continuous functions of bounded nth variation

1969 ◽  
Vol 16 (3) ◽  
pp. 205-214
Author(s):  
Gavin Brown
Keyword(s):  
Continuous Function ◽  
Positive Integer ◽  
Decomposition Theorem ◽  
Continuous Functions ◽  
Unit Interval ◽  
Jordan Decomposition ◽  
The Real ◽  
Elementary Construction ◽  
The Difference

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

scholarly journals The Köthe Dual of an Abstract Banach Lattice

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
E. Jiménez Fernández ◽  
M. A. Juan ◽  
E. A. Sánchez-Pérez
Keyword(s):  
Integral Representation ◽  
Banach Lattice ◽  
Broad Class ◽  
Banach Lattices ◽  
Vector Measures ◽  
Integrable Functions ◽  
Suitable Definition ◽  
Integral Structures ◽  
Definition Of ◽  
Köthe Dual

We analyze a suitable definition of Köthe dual for spaces of integrable functions with respect to vector measures defined onδ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Köthe dual with the topological dual is preserved.

scholarly journals An n-dimensional analogue of Cauchy's integral theorem

1960 ◽  
Vol 1 (2) ◽  
pp. 171-202 ◽  
Author(s):  
J. H. Michael
Keyword(s):  
Bounded Variation ◽  
Continuous Mapping ◽  
Topological Manifold ◽  
Parametric Surfaces ◽  
Integral Theorem ◽  
Cauchy’S Integral Theorem ◽  
Natural Way

As in [5] a parametric n-surface in Rk (where k ≧ n) will be a pair (f, Mn), consisting of a continuous mapping f of an oriented topological manifold Mn into the euclidean k-space Rk. (f, Mn) is said to be closed if Mn is compact. The main purpose of this paper is to use the method of [4] to prove a general form of Cauchy's Integral Theorem (Theorem 5.3) for those closed parametric n-surfaces (f, Mn) in Rn+1, which have bounded variation in the sense of [5] and for which f(Mn) has a finite Hausdorff n-measure. As in [4], the proof is carried out by approximating the surface with a simpler type of surface. However, when n > 1, a difficulty arises in that there are entities, which occur in a natural way, but are not parametric surfaces. We therefore introduce a concept which we call an S-system and which forms a generalisation (see 2.2) of the type of closed parametric n-surface that was studied in [5] II, 3 in connection with a proof of a Gauss-Green Theorem. The surfaces of [5] II, 3 include those that are studied in this paper.

scholarly journals The Jordan decomposition of bounded variation functions valued in vector spaces

2017 ◽  
Vol 2 (4) ◽  
pp. 635-646
Author(s):  
Francisco J. Mendoza-Torres ◽  
◽  
Juan A. Escamilla-Reyna ◽  
Daniela Rodríguez-Tzompantzi
Keyword(s):  
Bounded Variation ◽  
Vector Spaces ◽  
Jordan Decomposition ◽  
Bounded Variation Functions

scholarly journals Positive p-summing operators, vector measures and tensor products

1988 ◽  
Vol 31 (2) ◽  
pp. 179-184 ◽  
Author(s):  
Oscar Blasco
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Lattice ◽  
Tensor Products ◽  
Boundary Values ◽  
Vector Measures ◽  
Absolutely Summing Operators

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

A Lebesgue Decomposition Theorem for C* Algebras

1972 ◽  
Vol 15 (1) ◽  
pp. 87-91 ◽  
Author(s):  
Michael Henle
Keyword(s):  
Absolute Continuity ◽  
Linear Functional ◽  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
C Algebras ◽  
Positive Linear Functional ◽  
Decomposition Theorems

This paper, by generalizing von Neumann's proof of the Radon-Nikodym and Lebesgue decomposition theorems [3], obtains analogous results for positive linear functional on a C* algebra. The concept of "absolute continuity" used and the Radon-Nikodym portion of the resulting theorem are due to Dye [2].

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