On Integration of Vector-Valued Functions

1958 ◽  
Vol 10 ◽  
pp. 399-412 ◽  
Author(s):  
D. O. Snow
Keyword(s):  
Bochner Integral ◽  
Significant Extent ◽  
Vector Valued Functions ◽  
Integrating Vector ◽  
Vector Valued

Among the variety of integrals which have been devised for integrating vector-valued functions the most widely used is that of Bochner (2), perhaps because of the simplicity of its formulation. Other approaches, including one by Birkhoff (1), have yielded more general integrals yet none of them seems to have supplanted the Bochner integral to a significant extent.

scholarly journals Quadratic reverses of the triangle inequality for Bochner integral in Hilbert spaces

2004 ◽  
Vol 70 (3) ◽  
pp. 451-462
Author(s):  
Sever S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Triangle Inequality ◽  
Bochner Integral ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

Some quadratic reverses of the continuous triangle inequality for the Bochner integral of vector-valued functions in Hilbert spaces are given. Applications of complex-valued functions are provided as well.

REFINEMENTS OF THE CONTINUOUS TRIANGLE INEQUALITY FOR THE BOCHNER INTEGRAL IN HILBERT SPACES

2008 ◽  
Vol 01 (04) ◽  
pp. 521-533
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Triangle Inequality ◽  
Bochner Integral ◽  
Numerical Radius ◽  
Operator Inequalities ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

Some refinements of the continuous triangle inequality for the Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for norm and numerical radius operator inequalities are provided. A particular case of interest for complex-valued functions is pointed out as well.

On Measurability for Vector-Valued Functions

1963 ◽  
Vol 15 ◽  
pp. 613-621 ◽  
Author(s):  
D. O. Snow
Keyword(s):  
Banach Space ◽  
Infinite Series ◽  
Unconditional Convergence ◽  
Integration Theory ◽  
Bochner Integral ◽  
Norm Topology ◽  
Vector Valued Functions ◽  
Vector Valued

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.

scholarly journals Farkas-Type Results for Vector-Valued Functions with Applications

2017 ◽  
Vol 173 (2) ◽  
pp. 357-390 ◽  
Author(s):  
N. Dinh ◽  
M. A. Goberna ◽  
M. A. López ◽  
T. H. Mo
Keyword(s):  
Vector Valued Functions ◽  
Vector Valued

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

scholarly journals THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS

2014 ◽  
Vol 57 (1) ◽  
pp. 17-82 ◽  
Author(s):  
TUOMAS P. HYTÖNEN ◽  
ANTTI V. VÄHÄKANGAS
Keyword(s):  
Singular Integrals ◽  
Local Version ◽  
Square Functions ◽  
Property A ◽  
Martingale Differences ◽  
Umd Spaces ◽  
Function Lattice ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Similar Extension

AbstractWe extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

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