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1

Normal structure in modulated topological vector spaces

Kozlowski Wojciech M.

Commentationes Mathematicae

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2020

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Vol. 60, No. 1/2

1--11

EN

Following the author’s recent paper On modulated topological vector spaces and applications, Bull. Aust. Math. Soc. (2020), we discuss a notion of modulated topological vector spaces, that generalise, among others, Banach spaces and modular function spaces. The interest in modulars reflects the fact that the notions of “norm like” but “non-euclidean” (i.e., possibly without the triangle property and non-necessarily homogenous) constructs to measure a level of proximity between complex objects have been used extensively in statistics and applied in many empirical scientific projects requiring an objective differentiation between several classes of objects, efficiently applied in many modern clustering and Artificial Intelligence (AI) related computer algorithms. As an example of application, we prove some results, which extend fixed point theorems from the above mentioned paper, by moving from the setting of admissible sets to a simpler and more general setup, which covers also closed bounded sets. The theory of modulated topological vector spaces provides a very minimalistic framework, where powerful geometrical, fixed point, approximation and optimisation theorems are valid under a bare minimum of assumptions.

2

Fixed points of mappings in Klee admissible spaces

Górniewicz L., Ślosarski M.

Control and Cybernetics

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2007

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Vol. 36, no 3

825-832

EN

In this paper we generalize the Lefschetz fixed point theorem from the case of metric ANR-s to the case of acceptable subsets of Klee admissible spaces. The results presented in this paper were announced in an earlier publication of the authors.

3

Trans-separability in spaces of continuous vector-valued functions

Khan L. A.

Demonstratio Mathematica

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2004

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Vol. 37, nr 3

611--617

EN

Let X be a completely regular Hausdorff space and E a Hausdorff topological vector space (TVS). In this note, we study the notion of trans-separability for certain subspaces of Cb(X,E) endowed with the cr-compact-open and some related topologies.

4

Fourier analysis on locally convex spaces of distributions, II

Sinha R. P., Mohammed A. N.

Demonstratio Mathematica

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2003

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Vol. 36, nr 4

899--913

EN

This is the second in a series of papers, extending the theory of Fourier analysis to locally convex spaces of distributions (LCD-spaces). In this paper, LCD-spaces admitting conjugation and multiplier operators on LCD-spaces are discussed. It is also shown that if E is an LCD-space having C∞ as a dense subset, then E*, endowed with the topology of precompact convergence, is an LCD-space having C∞ as a dense subset.

5

Operator-valued multiiiiiplication operators on weighted function spaces

Khan L. A., Thaheem A. B.

Demonstratio Mathematica

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2002

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Vol. 35, nr 3

599-605

EN

Let X be a completely regular Hausdorff space, E a Hausdorff topological vector space, V a Nachbin family of weights on X, and CVb(X,E) the weighted space of continuous JS-valued functions on X. Let B(E) be the vector space of all continuous linear mappings from E into itself, endowed with the topology of uniform convergence on bounded sets. If phi: X -> B(E) is a continuous mapping and f zawiera CVb(X,E), let Mphi,(f) = phif, where (phif)(x) = (phi(x)(f(x) (x zawiera się X). In this paper we give a necessary and sufficient condition for Mphi to be the multiplication operator (i.e. continuous self-mapping) on CVb(X,E), where E is a general space or a locally bounded space. These results extend recent work of Singh and Manhas to a non-locally convex setting and that of the authors where phi has been considered to be a complex or E-valued map.

6

Fixed point theorems for admissible multivalued maps with nonconvex domains

Kim I.-S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 4

457-467

EN

Using the fixed point property of acyclic absolute neighbourhood retract due to L. Górniewicz, we prove that every compact admissible multivalued selfmap defined on an admissible pseudoconvex subset of a topological vector space has a fixed point.