We introduce the notion of uniform Fréchet differentiability of mappings between Banach spaces, and we give some sufficient conditions for this property to hold.

Let $ℳ$ be the Banach space of real measures on a $\sigma$-ring $\mathbf{R}$, let ${ℳ}^{\text{'}}$ be its dual, let $E$ be a quasi-complete locally convex space, let $E^{\text{'}}$ be its dual, and let $\mu$ be an $E$-valued measure on $\mathbf{R}$. If is shown that for any $\theta \in {ℳ}^{\text{'}}$ there exists an element $\int \theta d\mu$ of $E$ such that $⟨x^{\text{'}}\circ \mu ,\theta ⟩=⟨\int \theta d\mu ,x^{\text{'}}⟩$ for any $x^{\text{'}}\in E^{\text{'}}$ and that the map$$\theta \to \int \theta d\mu :{ℳ}^{\text{'}}\to E$$is order continuous. It follows that the closed convex hull of $\mu \left(\mathbf{R}\right)$ is weakly compact.

A Boolean algebra $𝒜$ has the interpolation property (property (I)) if given sequences $\left(a_n\right)$, $\left(b_m\right)$ in $𝒜$ with $a_n\le b_m$ for all $n,m$, there exists an element $b$ in $𝒜$ such that $a_n\le b\le b_n$ for all $n$. Let $𝒜$ denote an algebra with the property (I). It is shown that if $({\mu }_n:𝒜\to X)$ ($X$ a Banach space) is a sequence of strongly additive measures such that ${lim}_n{\mu }_n\left(a\right)$ exists for each $a\in 𝒜$, then $\mu \left(a\right)={lim}_n{\mu }_n\left(a\right)$ defines a strongly additive map from $𝒜$ to $X$and the ${\mu }_n^{\text{'}}s$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive $X$-valued measures defined...

Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction $\Gamma :X\to 2^{\Phi }$ to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.

The abstract Perron-Stieltjes integral in the Kurzweil-Henstock sense given via integral sums is used for defining convolutions of Banach space valued functions. Basic facts concerning integration are preseted, the properties of Stieltjes convolutions are studied and applied to obtain resolvents for renewal type Stieltjes convolution equations.

Let T be a kernel operator with values in a rearrangement invariant Banach function space X on [0,∞) and defined over simple functions on [0,∞) of bounded support. We identify the optimal domain for T (still with values in X) in terms of interpolation spaces, under appropriate conditions on the kernel and the space X. The techniques used are based on the relation between linear operators and vector measures.