Metric spaces
Authors
Description
The book was written to serve as a text for a course on Metric Spaces, as an introduction to Topology. The work was awarded the most important Brazilian literary award, the Jabuti, in 1978, in the Exact Sciences category. Students with notions of analysis will better understand the applications of the theory of metric spaces. Elon Lages Lima presents the basic language of topology, connected sets, limit, and complex and compact metric spaces.
Applications such as the fundamental theorem of algebra, the existence of continuous functions with no derivative at any point, the Peano curve, Picard’s theorem on the existence and uniqueness of solution for ordinary differential equations, Montel’s theorem related to normal families of analytic functions, the Stone–Weierstrass theorem, and Hilbert’s cube as universal separable space are described. These examples seek to show the strength and multifunctionality of theories.
Target audience
Higher education
Name: Metric spaces
Author(s): e Elon Lages Lima
Pages: 308
Publication: IMPA, 2020
ISBN: 978-65-990528-7-3
Edition: 6
1 Metric Spaces
1 Definition and examples of metric spaces
2 Balls and spheres
3 Limited sets
4 Distance from a Point to a Set
5 Isometries
6 Pseudo metrics
7 Exercises
2 Continuous Functions
1 Definition and examples
2 Elemental properties of continuous applications
3 Homeomorphisms
4 Equivalent metrics
5 Linear and multilinear transformations
6 Exercises
3 Basic Language of Topology
1 Open sets
2 Relations between open sets and continuity
3 Topological spaces
4 Closed sets
5 Exercises
4 Connected Sets
1 Definition and examples
2 General properties of connected sets
3 Connection by paths
4 Related components
5 Connection as a topological invariant
6 Exercises
5 Limits
1 Sequence limits
2 Real number sequences
3 Series
4 Convergence and topology
5 Sequences of functions
6 Infinite Cartesian products
7 Role limits
8 Exercises
6 Uniform Continuity
1 Observations and examples
2 Exercises
7 Full Metric Spaces
1 Cauchy sequences
2 Full metric spaces
3 Banach spaces and Hilbert spaces
4 Extension of Continuous Applications
5 Completing a Metric Space
6 Topologically complete metric spaces
7 Baire’s theorem
8 The method of successive approximations
9 Exercises
8 Compact Metric Spaces
1 Compactness on the straight
2 Compact Metric Spaces
3 Two-factor products, one of which is compact
4 A base for C ( K; M)
5 Characteristics of compact spaces
6 Cartesian products of compact spaces
7 Uniform continuity
8 Locally compact spaces
9 Finite-dimensional normed vector spaces
10 Equicontinuity
11 Weierstrass and Stone’s approximation theorems
12 Exercises
9 Breakaway Spaces
1 General properties
2 Locally compact separable spaces
3 The Hilbert cube as a universal separable space
4 The Hahn–Mazurkiewicz Theorem
5 Paracompacity
6 Exercises
Bibliography
Rating Index
Index
This book received the Jabuti Prize for Exact Sciences, granted by the Brazilian Book Chamber, in 1978