Riemann intergability versus continuity for vector-valued functions

2021 ◽  
Vol 30 (1) ◽  
pp. 49-60
Author(s):  
A. LONE NISAR ◽  
T. A. CHISHTI
Keyword(s):  
Continuous Functions ◽  
Interesting Topic ◽  
Weak Property ◽  
Modern Analysis ◽  
Vector Valued Functions ◽  
Vector Valued

The interplay between Riemann integrability and continuity is an interesting topic of modern analysis. In this paper, Riemann integrability of vector-valued continuous functions, property of Lebesgue and weak property of Lebesgue are surveyed and discussed. We also prove that `1(N, X) has the property of Lebesgue.

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.

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

Spaces of Continuous Vector Functions as Duals

1988 ◽  
Vol 31 (1) ◽  
pp. 70-78 ◽  
Author(s):  
Michael Cambern ◽  
Peter Greim
Keyword(s):  
Hilbert Space ◽  
Weak Topology ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Continuous Vector ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractA well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.

scholarly journals On approximation in weighted spaces of continuous vector-valued functions

1987 ◽  
Vol 29 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Liaqat Ali Khan
Keyword(s):  
Continuous Functions ◽  
Weighted Spaces ◽  
Regular Space ◽  
Local Convexity ◽  
Range Space ◽  
Completely Regular ◽  
Spaces Of Continuous Functions ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

Absolute Continuity of Some Vector Functions and Measures

1972 ◽  
Vol 24 (5) ◽  
pp. 737-746 ◽  
Author(s):  
William J. Knight
Keyword(s):  
Bounded Variation ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Continuous Point ◽  
Uniform Boundedness Principle ◽  
Vector Valued Functions ◽  
Analogous Theorem ◽  
Vector Valued

In the theory of vector valued functions there is a theorem which states that if a function from a compact interval I into a normed linear space X is of weak bounded variation, then it is of bounded variation. The proof uses in a straightforward way the Uniform Boundedness Principle (see [2, p. 60]). The present paper grew from the question of whether an analogous theorem holds for absolutely continuous functions. The answer is in the negative, and an example will be given (Theorem 7). But it will also be shown that if X is weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely continuous point function from / into X is absolutely continuous. The method of proof involves the construction of a countably additive set function in the standard Lebesgue-Stieltjes fashion.The paper is divided into three parts. In Section 1 extensions of finitely additive, absolutely continuous set functions are carried out in an abstract setting. Section 2 applies this to vector valued (point) functions on the real line.

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 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.

Sign in / Sign up

Export Citation Format

Share Document