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Curves in Banach spaces which allow a C 1 , BV parametrization or a parametrization with finite convexity

Jakub Duda, Luděk Zajíček (2013)

Czechoslovak Mathematical Journal

We give a complete characterization of those f : [ 0 , 1 ] X (where X is a Banach space) which allow an equivalent C 1 , 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 = n . We present examples which show applicability of our characterizations. For example, we show that the C 1 , BV and C 2 parametrization problems are equivalent for X = but are not equivalent for X = 2 .

Functions of class Ck without derivatives.

Gijs M. Tuynman (1997)

Publicacions Matemàtiques

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....

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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 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 X with respect to any norming subset there exists a separately increasing function g : [ 0 , 1 ] m such that the sets of points of discontinuity...

The weak McShane integral

Mohammed Saadoune, Redouane Sayyad (2014)

Czechoslovak Mathematical Journal

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ -finite outer regular quasi Radon measure space ( S , Σ , 𝒯 , μ ) 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...

Weaker forms of continuity and vector-valued Riemann integration

M. A. Sofi (2012)

Colloquium Mathematicae

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.

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