Let $\tilde{f}$, $\tilde{g}$ be ultradistributions in ${𝒵}^{\text{'}}$ and let ${\tilde{f}}_n=\tilde{f}*{\delta }_n$ and ${\tilde{g}}_n=\tilde{g}*{\sigma }_n$ where $\left\{{\delta }_n\right\}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $\delta$. Then the neutrix product $\tilde{f}\diamond \tilde{g}$ is defined on the space of ultradistributions ${𝒵}^{\text{'}}$ as the neutrix limit of the sequence $\left\{\frac{1}{2}({\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n)\right\}$ provided the limit $\tilde{h}$ exist in the sense that $$\underset{n\to \infty }{\mathrm{N}\text{-}lim}\frac{1}{2}\langle {\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n,\psi \rangle =\langle \tilde{h},\psi \rangle$$ for all $\psi$ in $𝒵$. We also prove that the neutrix convolution product $f♦*g$ exist in ${𝒟}^{\text{'}}$, if and only if the neutrix product $\tilde{f}\diamond \tilde{g}$ exist in ${𝒵}^{\text{'}}$ and the exchange formula $$F\left(f♦*g\right)=\tilde{f}\diamond \tilde{g}$$ is then satisfied.

In this paper, we give some necessary and sufficient conditions such that each positive operator between two Banach lattices is weak almost Dunford-Pettis, and we derive some interesting results about the weak Dunford-Pettis property in Banach lattices.

Some boundedness and VMO results are proved for a function f integrable on a cube $Q_0$, starting from an integral bound.

We show that the Lindenstrauss basic sequence in l₁ may be used to construct a conditional quasi-greedy basis of l₁, thus answering a question of Wojtaszczyk. We further show that the sequence of coefficient functionals for this basis is not quasi-greedy.

Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.

We construct a simplicial locally convex algebra, whose weak dual is the standard cosimplicial topological space. The construction is carried out in a purely categorical way, so that it can be used to construct (co)simplicial objects in a variety of categories --- in particular, the standard cosimplicial topological space can be produced.

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.

A constructive proof of the Beurling-Rudin theorem on the characterization of the closed ideals in the disk algebra A(𝔻) is given.

We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.

A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.

We prove the existence of a contractive mapping on a weakly compact convex set in a Banach space that is fixed point free. This answers a long-standing open question.