Integrals and Banach spaces for finite order distributions

Erik Talvila

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Displaying similar documents to “Integrals and Banach spaces for finite order distributions”

A general integral

R. Estrada, J. Vindas

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We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if f is locally distributionally integrable over the real line and ψ ∈ (ℝ) is a test function, then fψ is distributionally integrable, and the formula $⟨, ψ ⟩ = () \int_{- \infty}^{\infty} f (x) ψ (x) d x$ , defines a distribution...

Isomorphic and isometric copies of $ℓ_{\infty} (Γ)$ in duals of Banach spaces and Banach lattices

Marek Wójtowicz (2006)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $Γ$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^{*}$ contains an isometric copy of $c_{0}$ iff $X^{*}$ contains an isometric copy of $ℓ_{\infty}$ , and (2) $E^{*}$ contains a lattice-isometric copy of $c_{0} (Γ)$ iff $E^{*}$ contains a lattice-isometric copy of $ℓ_{\infty} (Γ)$ .

Role of the Harnack extension principle in the Kurzweil-Stieltjes integral

Umi Mahnuna Hanung (2024)

Mathematica Bohemica

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In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with...

On a generalization of Henstock-Kurzweil integrals

Jan Malý, Kristýna Kuncová (2019)

Mathematica Bohemica

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We study a scale of integrals on the real line motivated by the $M C_{α}$ integral by Ball and Preiss and some recent multidimensional constructions of integral. These integrals are non-absolutely convergent and contain the Henstock-Kurzweil integral. Most of the results are of comparison nature. Further, we show that our indefinite integrals are a.e. approximately differentiable. An example of approximate discontinuity of an indefinite integral is also presented.

Continuity in the Alexiewicz norm

Erik Talvila (2006)

Mathematica Bohemica

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If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $∥ f ∥ = {sup}_{I} | \int_{I} f |$ where the supremum is taken over all intervals $I \subset ℝ$ . Define the translation $τ_{x}$ by $τ_{x} f (y) = f (y - x)$ . Then $∥ τ_{x} f - f ∥$ tends to $0$ as $x$ tends to $0$ , i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $∥ τ_{x} f - f ∥$ can tend to 0 arbitrarily slowly. In general, $∥ τ_{x} f - f ∥ \geq osc f | x |$ as $x \to 0$ , where $osc f$ is the oscillation of $f$ . It is shown that if $F$ is a primitive of $f$ then $∥ τ_{x} F - F ∥ \leq ∥ f ∥ | x |$ . An example shows that the function $y \mapsto τ_{x} F (y) - F (y)$ need not be in $L^{1}$ . However, if...

Displaying similar documents to “Integrals and Banach spaces for finite order distributions”

A general integral

Isomorphic and isometric copies of ℓ ∞ ( Γ ) in duals of Banach spaces and Banach lattices

Role of the Harnack extension principle in the Kurzweil-Stieltjes integral

On a generalization of Henstock-Kurzweil integrals

Continuity in the Alexiewicz norm

Isomorphic and isometric copies of $ℓ_{\infty} (Γ)$ in duals of Banach spaces and Banach lattices