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Bayoumi quasi-differential is not different from Fréchet-differential

Fernando Albiac, José Ansorena (2012)

Open Mathematics

Unlike for Banach spaces, the differentiability of functions between infinite-dimensional nonlocally convex spaces has not yet been properly studied or understood. In a paper published in this Journal in 2006, Bayoumi claimed to have discovered a new notion of derivative that was more suitable for all F-spaces including the locally convex ones with a wider potential in analysis and applied mathematics than the Fréchet derivative. The aim of this short note is to dispel this misconception, since...

Coorbit space theory for quasi-Banach spaces

Holger Rauhut (2007)

Studia Mathematica

We generalize the classical coorbit space theory developed by Feichtinger and Gröchenig to quasi-Banach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasi-Banach spaces. These atomic decompositions are used to prove fast convergence rates of best n-term approximation schemes. We apply the abstract theory to time-frequency analysis of modulation spaces M m p , q , 0 < p,q ≤ ∞.

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ν ( X ) and ν ( X ) of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient conditions...

Diametral dimension of some pseudoconvex multiscale spaces

Jean-Marie Aubry, Françoise Bastin (2010)

Studia Mathematica

Stemming from the study of signals via wavelet coefficients, the spaces S ν are complete metrizable and separable topological vector spaces, parametrized by a function ν, whose elements are sequences indexed by a binary tree. Several papers were devoted to their basic topology; recently it was also shown that depending on ν, S ν may be locally convex, locally p-convex for some p > 0, or not at all, but under a minor condition these spaces are always pseudoconvex. We deal with some more sophisticated...

Ekeland's variational principle in locally p-convex spaces and related results

J. H. Qiu, S. Rolewicz (2008)

Studia Mathematica

In the framework of locally p-convex spaces, two versions of Ekeland's variational principle and two versions of Caristi's fixed point theorem are given. It is shown that the four results are mutually equivalent. Moreover, by using the local completeness theory, a p-drop theorem in locally p-convex spaces is proven.

Localization of bounded sets in tensor products.

A. Peris, M. J. Rivera (1996)

Revista Matemática de la Universidad Complutense de Madrid

The problem of topologies of Grothendieck is considered for complete tensor products of Fréchet spaces endowed with the topology defined by an arbitrary tensor norm. Some consequences on the stability of certain locally convex properties in spaces of operators are also given.

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