Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of $c_0$. Then the Bochner space $L^1(m;X)$ is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.

Let A be the class of all real-analytic functions and β the class of all Boehmians. We show that there is no continuous operation on β which is ordinary multiplication when restricted to A.

The fixed infinitely differentiable function $\rho \left(x\right)$ is such that $\left\{n\rho \right(nx\left)\right\}$ is a regular sequence converging to the Dirac delta function $\delta$. The function ${\delta }_{𝐧}\left(𝐱\right)$, with $𝐱=(x_1,\cdots ,x_m)$ is defined by $${\delta }_{𝐧}\left(𝐱\right)=n_1\rho \left(n_1{x}_1\right)\cdots n_m\rho \left(n_m{x}_m\right).$$ The product $f\circ g$ of two distributions $f$ and $g$ in ${𝒟}_m^{\text{'}}$ is the distribution $h$ defined by $$\underset{n_1\to \infty }{error}\cdots \underset{n_m\to \infty }{error}\langle f_{𝐧}{g}_{𝐧},\phi \rangle =\langle h,\phi \rangle ,$$ provided this neutrix limit exists for all $\phi \left(𝐱\right)={\phi }_1\left(x_1\right)\cdots {\phi }_m\left(x_m\right)$, where $f_{𝐧}=f*{\delta }_{𝐧}$ and $g_{𝐧}=g*{\delta }_{𝐧}$.

We study Palamodov's derived projective limit functor Proj¹ for projective spectra consisting of webbed locally convex spaces introduced by Wilde. This class contains almost all locally convex spaces appearing in analysis. We provide a natural characterization for the vanishing of Proj¹ which generalizes and unifies results of Palamodov and Retakh for spectra of Fréchet and (LB)-spaces. We thus obtain a general tool for solving surjectivity problems in analysis.

In this paper we discuss the problem of when the projective tensor product of two Banach spaces has the Radon-Nikodym property. We give a detailed exposition of the famous examples of Jean Bourgain and Gilles Pisier showing that there are Banach spaces X and Y such that each has the Radon-Nikodym property but for which their projective tensor product does not; this result depends on the classical theory of absolutely summing, integral and nuclear operators, as well as the famous Grothendieck inequality...

We study the properties of the weighted space $H_{2,\alpha }^k\left(\Omega \right)$ and weighted set $W_{2,\alpha }^k(\Omega ,\delta )$ for boundary value problem with singularity.

The main aim of this paper is to prove that a nuclear Fréchet space E has the property (Hu) (resp. (Ω)) if and only if every holomorphic function on E (resp. on some dense subspace of E) can be written in the exponential form.

A characterization of property $\left(\beta \right)$ of an arbitrary Banach space is given. Next it is proved that the Orlicz-Bochner sequence space $l_{\Phi }\left(X\right)$ has the property $\left(\beta \right)$ if and only if both spaces $l_{\Phi }$ and $X$ have it also. In particular the Lebesgue-Bochner sequence space $l_p\left(X\right)$ has the property $\left(\beta \right)$ iff $X$ has the property $\left(\beta \right)$. As a corollary we also obtain a theorem proved directly in [5] which states that in Orlicz sequence spaces equipped with the Luxemburg norm the property $\left(\beta \right)$, nearly uniform convexity, the drop property and...