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 $⟨,\psi ⟩=\left(\right){\int }_{-\infty }^{\infty }f\left(x\right)\psi \left(x\right)dx$, defines a distribution...

Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma$ 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 ${\ell }_{\infty }$, and (2) $E^{*}$ contains a lattice-isometric copy of $c_0\left(\Gamma \right)$ iff $E^{*}$ contains a lattice-isometric copy of ${\ell }_{\infty }\left(\Gamma \right)$.

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

In the paper [5] L. Drewnowski and the author proved that if $X$ is a Banach space containing a copy of $c_0$ then $L_1(\mu ,X)$ is complemented in $cabv(\mu ,X)$ and conjectured that the same result is true if $X$ is any Banach space without the Radon-Nikodym property. Recently, F. Freniche and L. Rodriguez-Piazza ([7]) disproved this conjecture, by showing that if $\mu$ is a finite measure and $X$ is a Banach lattice not containing copies of $c_0$, then $L_1(\mu ,X)$ is complemented in $cabv(\mu ,X)$. Here, we show that the complementability of $L_1(\mu ,X)$...

We prove some existence theorems for nonlinear integral equations of the Urysohn type $x\left(t\right)=\phi \left(t\right)+\lambda {\int }_0^{a}f(t,s,x\left(s\right))ds$ and Volterra type $x\left(t\right)=\phi \left(t\right)+{\int }_0^{t}f(t,s,x\left(s\right))ds$, $t\in I_a=[0,a]$, where f and φ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently $B({\ell }_{\infty },X*)=\Pi ₂({\ell }_{\infty },X*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could...