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## Über dieses Buch

Aimed toward researchers and graduate students familiar with elements of functional analysis, linear algebra, and general topology; this book contains a general study of modulars, modular spaces, and metric modular spaces. Modulars may be thought of as generalized velocity fields and serve two important purposes: generate metric spaces in a unified manner and provide a weaker convergence, the modular convergence, whose topology is non-metrizable in general. Metric modular spaces are extensions of metric spaces, metric linear spaces, and classical modular linear spaces. The topics covered include the classification of modulars, metrizability of modular spaces, modular transforms and duality between modular spaces, metric and modular topologies. Applications illustrated in this book include: the description of superposition operators acting in modular spaces, the existence of regular selections of set-valued mappings, new interpretations of spaces of Lipschitzian and absolutely continuous mappings, the existence of solutions to ordinary differential equations in Banach spaces with rapidly varying right-hand sides.

## Inhaltsverzeichnis

### Chapter 1. Classes of Modulars

Abstract
In this chapter, we study variants of modular axioms and transformations of modulars, which preserve the modularity property. It is shown that these transforms are more flexible than the metric transforms.
Vyacheslav V. Chistyakov

### Chapter 2. Metrics on Modular Spaces

Abstract
In this chapter, we address the metrizability of modular spaces.
Vyacheslav V. Chistyakov

### Chapter 3. Modular Transforms

Abstract
In this chapter, we study variants of modular axioms and transformations of modulars, which preserve the modularity property. It is shown that these transforms are more flexible than the metric transforms.
Vyacheslav V. Chistyakov

### Chapter 4. Topologies on Modular Spaces

Abstract
In this chapter, we introduce, study and compare two kinds of convergences and topologies, induced by a pseudomodular on a set—metric and modular.
Vyacheslav V. Chistyakov

### Chapter 5. Bounded and Regulated Mappings

Abstract
In this chapter, we introduce and study a special $$\mathbb{N}$$-valued modular on the set of all mappings from an interval of the real line into a metric space. We show that the sets of all bounded mappings and regulated mappings (i.e., those, whose one-sided limits exist at each point of the interval) are modular spaces for this modular. We apply the modular to establish a pointwise selection principle, extending the classical Helly Selection Theorem.
Vyacheslav V. Chistyakov

### Chapter 6. Mappings of Bounded Generalized Variation

Abstract
Here we follow the notation of Chap. 5, and denote the pseudomodular from Definition 5.1.1 more precisely by $$w_{\lambda }^{\mathbb{N}}(x,y)$$. For I = [a, b] and a metric space (M, d), we define new pseudomodulars on the set X = M I , whose induced modular spaces consist of mappings of bounded generalized variation (in the sense of Jordan, Wiener-Young, Riesz-Medvedev). We prove the Lipschitz continuity of a superposition operator (of “multiplication”) and establish the existence of selections of bounded variation of compact-valued BV multifunctions. An application to ordinary differential equations in Banach spaces is also given.
Vyacheslav V. Chistyakov

### Backmatter

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