### A constructive integral equivalent to the integral of Kurzweil

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

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

We give a complete characterization of those $f:[0,1]\to X$ (where $X$ is a Banach space) which allow an equivalent ${C}^{1,\mathrm{BV}}$ parametrization (i.e., a ${C}^{1}$ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for $X={\mathbb{R}}^{n}$. We present examples which show applicability of our characterizations. For example, we show that the ${C}^{1,\mathrm{BV}}$ and ${C}^{2}$ parametrization problems are equivalent for $X=\mathbb{R}$ but are not equivalent for $X={\mathbb{R}}^{2}$.

We describe a general axiomatic way to define functions of class Ck, k ∈ N∪{∞} on topological abelian groups. In the category of Banach spaces, this definition coincides with the usual one. The advantage of this axiomatic approach is that one can dispense with the notion of norms and limit procedures. The disadvantage is that one looses the derivative, which is replaced by a local linearizing factor. As an application we use this approach to define C∞ functions in the setting of graded/super manifolds....

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f:{[0,1]}^{m}\to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f:{[0,1]}^{m}\to X$ with respect to any norming subset there exists a separately increasing function $g:{[0,1]}^{m}\to \mathbb{R}$ such that the sets of points of discontinuity...

The McShane and Kurzweil-Henstock integrals for functions taking values in a locally convex space are defined and the relations with other integrals are studied. A characterization of locally convex spaces in which Henstock Lemma holds is given.

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a $\sigma $-finite outer regular quasi Radon measure space $(S,\Sigma ,\mathcal{T},\mu )$ into a Banach space $X$ and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function $f$ from $S$ into $X$ is weakly McShane integrable on each measurable subset of $S$ if and only if...

It was proved by Kadets that a weak*-continuous function on [0,1] taking values in the dual of a Banach space X is Riemann-integrable precisely when X is finite-dimensional. In this note, we prove a Fréchet-space analogue of this result by showing that the Riemann integrability holds exactly when the underlying Fréchet space is Montel.