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Published in:
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2018 | OriginalPaper | Chapter

1. Metric and Topological Spaces

Author : Vladimir Kadets

Published in: A Course in Functional Analysis and Measure Theory

Publisher: Springer International Publishing

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Abstract

Topological, and especially metric spaces, are frequently mentioned and used in courses on mathematical analysis, linear algebra (in which one of the most important examples of metric spaces, the finite-dimensional Euclidean space, is examined), differential geometry (in which geodesic curves and the intrinsic metric of a surface are studied), and also, it goes without saying, in topology courses. For this reason, we only briefly recall the well-known definitions and facts, discuss the terminology and notation adopted in this book, while dwelling in more detail on issues that possibly are not treated in other courses.

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next chapter Measure Theory
Footnotes
1
“... and the other explained to me that inside the globe there was another globe much bigger than the outer one” (Jaroslav Hašek, “The good solider Švejk”, Chapter “Švejk thrown out of the lunatic asylum”).
 
2
In this definition the symbol \(\rho \) is used with two distinct meanings: on the left — as the distance in \(\ell _\infty (\Gamma , X)\), and on the right — as the distance in X. This ambiguity can be removed by denoting the metric on the space X by \(\rho _X\).
 
3
The isoperimetric problem is associated with the legend of queen Dido, the founder of Carthage (in modern-day Tunisia). When the colonists arrived in the new place, the locals did not receive them kindly. The request of allotting a piece of land for the construction of the new city was de facto rejected by the following statement: “You are allowed to use as much land as can be encompassed by the skin (hide) of an ox.” However, Dido did not allow herself to get confused and faced the challenge. She ordered her people to cut the skin into very thin strips and, connecting them, mark the border of the new settlement. Needless to say, in doing so it was desirable to obtain the largest area, i.e., to solve the isoperimetric problem.
 
Metadata
Title
Metric and Topological Spaces
Author
Vladimir Kadets
Copyright Year
2018
Book
A Course in Functional Analysis and Measure Theory

Print ISBN: 978-3-319-92003-0

Electronic ISBN: 978-3-319-92004-7

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-92004-7_1

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