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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 lifting theorem for locally convex subspaces of L 0

R. Faber (1995)

Studia Mathematica

We prove that for every closed locally convex subspace E of L 0 and for any continuous linear operator T from L 0 to L 0 / E there is a continuous linear operator S from L 0 to L 0 such that T = QS where Q is the quotient map from L 0 to L 0 / E .

A Lipschitz function which is C ∞ on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is C ∞ smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially smooth...

A new proof of Fréchet differentiability of Lipschitz functions

Joram Lindenstrauss, David Preiss (2000)

Journal of the European Mathematical Society

We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on ℓ 2 (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w * closure of the set of its points of Gâteaux differentiability is norm separable.

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 a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form F ( x ) : = F ̃ x ( 2 n + 1 ) where F ̃ : X 2 n + 1 → Y is a continuous (2n+1)-linear operator.

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