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## Über dieses Buch

This book presents a detailed, self-contained theory of continuous mappings. It is mainly addressed to students who have already studied these mappings in the setting of metric spaces, as well as multidimensional differential calculus. The needed background facts about sets, metric spaces and linear algebra are developed in detail, so as to provide a seamless transition between students' previous studies and new material.
In view of its many novel features, this book will be of interest also to mature readers who have studied continuous mappings from the subject's classical texts and wish to become acquainted with a new approach. The theory of continuous mappings serves as infrastructure for more specialized mathematical theories like differential equations, integral equations, operator theory, dynamical systems, global analysis, topological groups, topological rings and many more. In light of the centrality of the topic, a book of this kind fits a variety of applications, especially those that contribute to a better understanding of functional analysis, towards establishing an efficient setting for its pursuit.

## Inhaltsverzeichnis

### Chapter 1. Overview

Abstract
Continuity theory serves as infrastructure for more specialized mathematical theories such as ordinary differential equations, partial differential equations, integral equations, operator theory, dynamical systems, and global analysis. The infrastructure includes creation of new spaces out of given ones, extension theorems, existence theorems, inversion theorems, approximation theorems, factorization theorems, adjunctions (e.g., exponential laws), and (local) representation theorems. So it embodies a great variety of possible topics. The present book, while deliberately not encyclopedic, does include a systematic study of linear continuity—enough to provide a foundation for functional analysis (the linear part of continuity theory).
Louis Nel

### Chapter 2. General Preparation

Abstract
This chapter sets forth prerequisite knowledge required for the following chapters. Where you are already familiar with something, just attend to the notation and terminology.
Louis Nel

### Chapter 3. Continuity Enabling Structures

Abstract
Continuous mappings are defined via functions that preserve convergence. In this chapter we discuss selected classes of spaces that makes the definition of convergence possible and therefore also the definition of continuous mapping. There is no attempt to be encyclopedic. It will be seen that for every kind of space there are certain auxiliary concepts that enable expression of continuity in various equivalent ways. The filter concept is one such auxiliary. We introduce it even for real intervals, where sequential convergence is adequate, but where students can get used to it in a familiar environment.
Louis Nel

### Chapter 4. Construction of New Spaces

Abstract
We begin this section with the construction of a cartesian product for a given set of C-spaces. It serves as a model for many similar constructions.
Louis Nel

### Chapter 5. Various Kinds of Spaces

Abstract
In this chapter we study various spatial properties. When a C-space has a property P, several things are of interest: alternative conditions that implies P, what is implied by P and what constructions will preserve P. Then, of course, there is the separate issue of the class of spaces that have property P: what constructions are available in this class.
Louis Nel

### Chapter 6. Fundamentals of Linear Continuity

Abstract
Functional analysis uses mappings which are simultaneously continuous and linear. This calls for a blend of convergence and vector structures. Linear continuity adds significant insight and perspective to the study of continuous mappings. It is a great source of nontrivial examples of continuous mappings between infinite dimensional spaces. It will be seen in a later chapter to reveal remarkable properties of all continuous mappings while being also of considerable intrinsic interest in its own right.
Louis Nel

### Chapter 7. Basic Categorical Concepts

Abstract
We have encountered, among others, the classes S (functions between sets), C (continuous mappings between convergence spaces), V (linear mappings between vector spaces), and CV. Members of these classes are triples of the form (domain, graph, and codomain). We have also encountered other classes of triples, e.g., the triples (x, ≤ , y) that arise in an up-directed set. The concept of category recognizes and exploits certain common features of such classes of triples of sets. For the sake of common generalization it introduces neutral terminology: the term “object” to represent “set” or “space” among possible choices; the term “arrow” or “morphism” to represent “function, mapping, homomorphism or relation” among possible choices.
Louis Nel

### Chapter 8. The Category C

Abstract
The category C has remarkable properties that make it particularly suitable to serve as foundation for a theory of continuous mappings. This chapter gives a systematic account of these properties.
Louis Nel

### Chapter 9. Reflective Subcategories of C

Abstract
Despite its impressive qualifications, the foundational category C (or one of its rigid-reflective alternatives C r and C p ) cannot by itself be the ultimate laboratory for continuity theory. Being a foundational category, it is inevitably infested with pathological spaces. We want to get rid of them while retaining the desirable properties of the category as a whole. By forming a reflective subcategory we automatically retain dicompleteness, thus also canonical factorizations. By forming an enriched reflective subcategory we retain poweredness along with dicompleteness.
Louis Nel

### Chapter 10. Enriched Dualities

Abstract
The classical Gelfand-Naimark duality expresses dual equivalence of the category of compact spaces and the category of rings of continuous $$\mathbb{R}$$-valued mappings on these spaces. The classical Stone duality expresses dual equivalence of the category of Stone spaces (compact zero-dimensional) and the category of Boolean rings of continuous mappings on Stone spaces. In this chapter we set forth representations that are, on the one hand, reminiscent of these long known classical dualities while, on the other hand, significantly different.
Louis Nel

### Chapter 11. The Category CV

Abstract
In this chapter we resume the study of CV as two-fold concrete category that was commenced in Sect. 10.​1 We show it to be dicomplete, how the limits and colimits are formed and how they relate to limits and colimits in the underlying categories. Then we show that CV has parapower-derived powers [E, F], paratensor products XE, and tensor products EF, all of which uphold appropriate exponential laws. The categorical nature of these developments make them applicable to any foundational category in the role of C.
Louis Nel

### Chapter 12. Reflective Subcategories of CV

Abstract
Much like C, the dicomplete category CV has all spaces that we want to be there but also some unwelcome ones. So to upgrade the quality of spaces without disturbing the good quality of the category, we proceed somewhat as we did for C, but with some difference.
Louis Nel

### Chapter 13. Linear Continuous Representations

Abstract
This chapter deals with a variety of linear continuous representations. It begins with representation of the gauged reflection of a paradual $$\mathtt{C}\,X = \mathsf{C}[X, \mathbb{K}]$$. This gives valuable insight into the nature of CV-functionals on CX. Further valuable insight comes from a representation of CV-functionals on CQ with compact Q via free CV-functionals. These results pave the way for a (new) proof that every CX is reflexive. This again leads to the noteworthy result that all cGV-spaces (complete locally convex topological vector spaces) are reflexive, whence cGV is dually equivalent to a category in which all spaces are complete and locally compact. Then, elaborating on the preliminary representation via free functionals, we derive a Riesz-Radon representation of $$\Delta \mathtt{C}\,X$$ for all C-spaces X, thus generalizing the earlier representation obtained for compact X.
Louis Nel

### Chapter 14. Smooth Continuity

Abstract
This final chapter will serve to provide further illustration of categorical methods applied to the category oCV. It does so by initiating an infinite dimensional differentiation theory of interest in its own right.
Louis Nel

### Backmatter

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