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A Borel extension approach to weakly compact operators on C 0 ( T )

Thiruvaiyaru V. Panchapagesan (2002)

Czechoslovak Mathematical Journal

Let X be a quasicomplete locally convex Hausdorff space. Let T be a locally compact Hausdorff space and let C 0 ( T ) = { f T I , f is continuous and vanishes at infinity } be endowed with the supremum norm. Starting with the Borel extension theorem for X -valued σ -additive Baire measures on T , an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map u C 0 ( T ) X to be weakly compact.

A full characterization of multipliers for the strong ρ -integral in the euclidean space

Lee Tuo-Yeong (2004)

Czechoslovak Mathematical Journal

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong ρ -integral, introduced by Jarník and Kurzweil. Let ( 𝒮 ρ ( E ) , · ) be the space of all strongly ρ -integrable functions on a multidimensional compact interval E , equipped with the Alexiewicz norm · . We show that each element in the dual space of ( 𝒮 ρ ( E ) , · ) can be represented as a strong ρ -integral. Consequently, we prove that f g is strongly ρ -integrable on E for each strongly ρ -integrable function f if and only if g is almost everywhere...

A generalized Pettis measurability criterion and integration of vector functions

I. Dobrakov, T. V. Panchapagesan (2004)

Studia Mathematica

For Banach-space-valued functions, the concepts of 𝒫-measurability, λ-measurability and m-measurability are defined, where 𝒫 is a δ-ring of subsets of a nonvoid set T, λ is a σ-subadditive submeasure on σ(𝒫) and m is an operator-valued measure on 𝒫. Various characterizations are given for 𝒫-measurable (resp. λ-measurable, m-measurable) vector functions on T. Using them and other auxiliary results proved here, the basic theorems of [6] are rigorously established.

A new relationship between decomposability and convexity

Bianca Satco (2006)

Commentationes Mathematicae Universitatis Carolinae

In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm · (where f = sup [ a , b ] [ 0 , 1 ] a b f ( s ) d s ) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.

A note on copies of c 0 in spaces of weak* measurable functions

Juan Carlos Ferrando (2000)

Commentationes Mathematicae Universitatis Carolinae

If ( Ω , Σ , μ ) is a finite measure space and X a Banach space, in this note we show that L w * 1 ( μ , X * ) , the Banach space of all classes of weak* equivalent X * -valued weak* measurable functions f defined on Ω such that f ( ω ) g ( ω ) a.e. for some g L 1 ( μ ) equipped with its usual norm, contains a copy of c 0 if and only if X * contains a copy of c 0 .

A note on the construction of measures taking their values in a Banach space with basis.

María Congost Iglesias (1983)

Stochastica

If E is a Banach space with a basis {en}, n belonging to N, a vector measure m: a --> E determines a sequence {mn}, n belonging to N, of scalar measures on a named its components. We obtain necessary and sufficient conditions to ensure that when given a sequence of scalar measures it is possible to construct a vector valued measure whose components were those given. Furthermore we study some relations between the variation of the measure m and the variation of its components.

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