We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by ${\sum }_{n=1}^{\infty }{x}_n{\chi }_{E_n}$, where $x_n$ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series ${\sum }_{n=1}^{\infty }{x}_n\left|E_n\right|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability...

A Banach space $X$ has the reciprocal Dunford-Pettis property ($RDPP$) if every completely continuous operator $T$ from $X$ to any Banach space $Y$ is weakly compact. A Banach space $X$ has the $RDPP$ (resp. property $\left(wL\right)$) if every $L$-subset of $X^{*}$ is relatively weakly compact (resp. weakly precompact). We prove that the projective tensor product $X\otimes {}_{\pi }Y$ has property $\left(wL\right)$ when $X$ has the $RDPP$, $Y$ has property $\left(wL\right)$, and $L(X,Y^{*})=K(X,Y^{*})$.

The space $\mathrm{\mathscr{H}𝒦}$ of Henstock-Kurzweil integrable functions on $[a,b]$ is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. In this paper, on each Fréchet space $\mathrm{\mathscr{H}𝒦}\left(X\right)$, an $F$-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is known that every control-convergent sequence in the $\mathrm{\mathscr{H}𝒦}$ space always belongs to a $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space for some $X$. We illustrate how...

Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_n$ and the corresponding primitive $F_n$. The pointwise convergence of the integrands $f_n$ to some $f$ and the equiintegrability of the functions $f_n$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_n$ converge uniformly to $F$. In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.