In this paper it is proved that if $\{E_n{\}}_{n=1}^{\infty }$ and $\{F_n{\}}_{n=1}^{\infty }$ are two sequences of infinite-dimensional Banach spaces then $H=\left({\oplus }_{n=1}^{\infty }{E}_n\right)\times {\prod }_{n=1}^{\infty }{F}_n$ is not $B_r$-complete. If $\{E_n{\}}_{n=1}^{\infty }$ and $\{F_n{\}}_{n=1}^{\infty }$ are also reflexive spaces there is on $H$ a separated locally convex topology $ℱ$, coarser than the initial one, such that $H\left[ℱ\right]$ is a bornological barrelled space which is not an inductive limit of Baire spaces. It is given also another results on $B_r$-completeness and bornological spaces.

It is proved that if ${\phi }_n$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form ${\sum }_{k=1}^{\infty }{c}_k{\phi }_k\left(x\right)$, where $c_k\in l_q$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f\in L_{\mu }^p[0,1]=f:{ʃ}_0^1{\left|f\left(x\right)\right|}^p\mu \left(x\right)dx<\infty$ there are numbers ${\varepsilon }_k$, k=1,2,…, ${\varepsilon }_k$ = 1 or 0, such that $li{m}_{n\to \infty }{ʃ}_0^1{|{\sum }_{k=1}^n{\varepsilon }_k{c}_k{\phi }_k\left(x\right)-f\left(x\right)|}^p\mu \left(x\right)dx=0.$ 2. For every p ∈ [1,2) and $f\in L_{\mu }^p[0,1]$ there are a function $g\in L^1[0,1]$ with g(x) = f(x) on E and numbers ${\delta }_k$, k=1,2,…, ${\delta }_k=1$ or 0,...

Let $P(\lambda ,D)={\sum }_{\left|\alpha \right|\le m}{a}_{\alpha }\left(\lambda \right)D^{\alpha }$ be a differential operator with constant coefficients $a_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\{$ P($\lambda$,D)$\}$ is of constant strength. We investigate the equation $P(\lambda ,D){𝔣}_{\lambda }\equiv g_{\lambda }$ where ${𝔤}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${𝔣}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander. ...

Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^{*}$ of a Banach space X, then ${\ell }^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of ${\ell }^1$ contains a copy of ${\ell }^1$ that is complemented in ${\ell }^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1[0,1]$ contains a copy of ${\ell }^1$ that is complemented in $L^1[0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of...

This article is devoted to a study of locally convex topologies on $\mathbf{H}\left(U\right)$ (where $U$ is an open subset of the locally convex topological vector space $E$ and $\mathbf{H}\left(U\right)$ is the set of all complex valued holomorphic functions on $E$). We discuss the following topologies on $\mathbf{H}\left(U\right)$: (a) the compact open topology ${\mathbf{I}}_0$, (b) the bornological topology associated with ${\mathbf{I}}_0$, (c) the ported topology of Nachbin ${\mathbf{I}}_{\omega }$, (d) the bornological topology associated with ${\mathbf{I}}_{\omega }$ ; and ...

Let $𝒜$ be a $\sigma$-algebra on a set $X$. If $A$ belongs to $𝒜$ let $e\left(A\right)$ be the characteristic function of $A$. Let ${\ell }_0^{\infty }(X,𝒜$ be the linear space generated by $\left\{e\right(A):A\in 𝒜\}$ endowed with the topology of the uniform convergence. It is proved in this paper that if $\left(E_n\right)$ is an increasing sequence of subspaces of ${\ell }_0^{\infty }(X,𝒜)$ covering it, there is a positive integer $p$ such that $E_p$ is a dense barrelled subspace of ${\ell }_0^{\infty }(X,𝒜)$, and some new results in measure theory are deduced from this fact.

For every element x** in the double dual of a separable Banach space X there exists the sequence $\left(x^{\left(2n\right)}\right)$ of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class $B_1\left(X\right)╲B_{1/2}\left(X\right)$ (resp. to the class $B_{1/4}\left(X\right)$) as the elements with the sequence $\left(x^{\left(2n\right)}\right)$ equivalent to the usual basis of ${\ell }^1$ (resp. as the elements with the sequence $(x^{(4n-2)}-x^{\left(4n\right)})$ equivalent to the...