### A bornological approach to rotundity and smoothness applied to approximation.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Modificando adecuadamente el método de un trabajo olvidado [1], probamos que si una aplicación continua, de un subconjunto abierto no vacío U de un espacio vectorial topológico metrizable separable y de Baire E, en un espacio localmente convexo, es direccionalmente diferenciable por la derecha en U según un subconjunto comagro de E, entonces, es genéricamente Gâteaux diferenciable en U. Nuestro resultado implica que cualquier espacio vectorial topológico, metrizable, separable y de Baire, es débilmente...

Many authors have recently studied compact and weakly compact homomorphisms between function algebras. Among them, Lindström and Llavona [2] treat weakly compact continuous homomorphisms between algebras of type C(T) when T is a completely regular Hausdorff space.Llavona asked wether the results in [2] are valid in the case of algebras of differentiable functions on Banach spaces. The purpose of this note is to give an affirmative answer to this question, by proving that weakly compact homomorphisms...

Using a construction similar to an iterated function system, but with functions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description of a trivial analytic multifunction.

In the paper a class of families (M) of functions defined on differentiable manifolds M with the following properties: ${1}_{}$. if M is a linear manifold, then (M) contains convex functions, ${2}_{}$. (·) is invariant under diffeomorphisms, ${3}_{}$. each f ∈ (M) is differentiable on a dense ${G}_{\delta}$-set, is investigated.

We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.

Let Ω be an open subset of a real Banach space E and, for 1 ≤ m ≤, let Cm(Ω) denote the algebra of all m-times continuously Fréchet differentiable real functions defined on Ω. We are concerned here with the question as to wether every nonzero algebra homomorphism φ: Cm(Ω) → R is given by evaluation at some point of Ω, i.e., if there exists some a ∈ Ω such that φ(f) = f(a) for each f ∈ Cm(Ω). This problem has been considered in [1,4,5] and [6]. In [6], a positive answer is given in the case that...

We observe that each set from the system $\tilde{\mathcal{A}}$ (or even $\tilde{\mathcal{C}}$) is $\Gamma $-null; consequently, the version of Rademacher’s theorem (on Gâteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on ${\mathbb{R}}^{n}$ is $\sigma $-strongly lower porous. A discussion concerning sets of Fréchet non-differentiability points of continuous convex...

Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $\Gamma :X\to {2}^{\Phi}$ be a cyclic ${\Phi}^{\gamma (\xb7,\xb7)}$-monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the ${\Phi}^{\gamma (\xb7,\xb7)}$-subdifferential of f, $\Gamma \left(x\right)\subset {\partial}_{\Phi}^{\gamma (\xb7,\xb7)}{f|}_{x}$.

In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.