scholarly journals Remarks on the semivariation of vector measures with respect to Banach spaces.

2007 ◽  
Vol 75 (3) ◽  
pp. 469-480 ◽  
Author(s):  
Oscar Blasco
Keyword(s):  
Banach Spaces ◽  
Vector Measures ◽  
Tensor Norm

Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.

Vector Integration and Stochastic Integration in Banach Spaces

2018 ◽  
Author(s):  
Nicolae Dinculeanu
Keyword(s):  
Banach Spaces ◽  
Positive Measure ◽  
Integration Theory ◽  
Stochastic Integration ◽  
Bochner Integral ◽  
Finite Variation ◽  
Vector Measures ◽  
Vector Integration ◽  
Step Functions ◽  
Integrable Variation

This article deals with vector integration and stochastic integration in Banach spaces. In particular, it considers the theory of integration with respect to vector measures with finite semivariation and its applications. This theory reduces to integration with respect to vector measures with finite variation which, in turn, reduces to the Bochner integral with respect to a positive measure. The article describes the four stages in the development of integration theory. It first provides an overview of the relevant notation for Banach spaces, measurable functions, the integral of step functions, and measurability with respect to a positive measure before discussing the Bochner integral. It then examines integration with respect to measures with finite variation, semivariation of vector measures, integration with respect to a measure with finite semivariation, and stochastic integrals. It also reviews processes with integrable variation or integrable semivariation and concludes with an analysis of martingales.

scholarly journals On copies of the null sequence Banach space in some vector measure spaces

1999 ◽  
Vol 59 (3) ◽  
pp. 443-447
Author(s):  
J.C. Ferrando ◽  
J.M. Amigó
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Vector Measure ◽  
Null Sequence ◽  
Vector Measures ◽  
Measure Spaces ◽  
Additive Vector

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.

Non-complemented Spaces of Operators, Vector Measures, and co

2012 ◽  
Vol 55 (3) ◽  
pp. 548-554 ◽  
Author(s):  
Paul Lewis ◽  
Polly Schulle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Compact Operators ◽  
Vector Measures ◽  
Spaces Of Operators ◽  
Space Of Compact Operators

AbstractThe Banach spaces L(X, Y), K(X,Y), Lw* (X*, Y), and Kw* (X*, Y) are studied to determine when they contain the classical Banach spaces co or ℓ∞. The complementation of the Banach space K(X, Y) in L(X, Y) is discussed as well as what impact this complementation has on the embedding of co or ℓ∞ in K(X, Y) or L(X, Y). Results of Kalton, Feder, and Emmanuele concerning the complementation of K(X, Y) in L(X, Y) are generalized. Results concerning the complementation of the Banach space Kw* (X*, Y) in Lw* (X*, Y) are also explored as well as how that complementation affects the embedding of co or ℓ∞ in Kw* (X*, Y) or Lw* (X*, Y). The ℓp spaces for 1 = p < ∞ are studied to determine when the space of compact operators from one ℓp space to another contains co. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.

scholarly journals On products of vector measures

1975 ◽  
Vol 19 (1) ◽  
pp. 91-96 ◽  
Author(s):  
U. K. Bandyopadhyay
Keyword(s):  
Banach Spaces ◽  
Convergence Theorem ◽  
Classical Result ◽  
Fubini Theorem ◽  
Bochner Integral ◽  
Vector Measures ◽  
Vector Integration ◽  
Two Measures ◽  
Dominated Convergence

Products of positive measures play a very important role in analysis. The purpose of this paper is to construct a theory of products of two measures taking values in two (possibly different) Banach spaces. A Fubini theorem is obtained which generalizes the Fubinitheorem for the Bochner integral (Dunford and Schwartz (1958), Theorem 9, page 190), and hence also the classical result.We use the theory of vector integration presented in Dinculeanu (1967). Our arguments rely upon a standard sort of application of the dominated convergence theorem (cf. Dunford and Schwartz (1958), Theorem 9, page 190), and therefore do not appear to generalize to any theory of integration where this theorem is lacking (e.g. Bartle (1956)).

Sign in / Sign up

Export Citation Format

Share Document