A concrete approach to classical analysis

443 Pages·2009·4.442 MB·English

by  Marian Muresan

CMS Books in Mathematics
Marian Mures¸an
A Concrete
Approach to
Classical Analysis
Canadian Mathematical Society
Société mathématique du Canada
Canadian Mathematical Society
Ouvrages de mathe´matiques de la SMC
Jonathan Borwein
Peter Borwein

Marian Mures¸an
A Concrete Approach
to Classical Analysis
ABC
MarianMures¸an
FacultyofMathematics&ComputerScience
Babes¸-BolyaiUniversity
Str.MihailKogalniceanunr1.
400084Cluj-Napoca
Romania
[email protected]
ISBN978-0-387-78932-3 e-ISBN978-0-387-78933-0
DOI 10.10079/78-0-387-78933-0
LibraryofCongressControlNumber:2008939066
MathematicsSubjectClassification(2000):46-xx
(cid:1)c SpringerScience+BusinessMedia,LLC2 009
Allrights reserved.Thisworkmaynotbetranslated orcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY
10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection
withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
9 8 7 6 5 4 3 2 1
springer.com
To the memory of my mother

vii
Computo ergo sum
D. H. Bailey1
E´ν o˜ιδα o´τι δ(cid:6)´ν o˜ιδα
(The only thing I know is that I don’t know anything)
Socrates2
Also, he made a molten sea of ten cubits from brim to brim,
round in compass, and five cubits the height thereof;
and a line of thirty cubits did compass it round about.
Old Testament, 1 Kings 7:23
We would be amiss, however, if we did not emphasize that the
extended precision calculation of π has substantial application
as a test of the “global integrity” of a supercomputer.
D. H. Bailey, J. M. Borwein,3 and P. B. Borwein4 in [10]
...the main goal of computation in pure mathematics is
arguably to yield insight.
[22]
1 David H. Bailey, Chief Technologist, Computational Research Dept., Lawrence
Berkeley Laboratory.
2 Socrates of Athens (Σωκρατης o Aϑηναιoς) ∼470–∼399 (BC).
3 Jonathan Michael Borwein, 1951–.
4 Peter Benjamin Borwein, 1953–.

Contents
1 Sets and Numbers ......................................... 1
1.1 Sets ................................................... 1
1.1.1 The concept of a set ............................... 1
1.1.2 Operations on sets................................. 3
1.1.3 Relations and functions ............................ 6
1.2 Sets of numbers ......................................... 11
1.2.1 Two examples .................................... 11
1.2.2 The real number system............................ 12
1.2.3 Elements of algebra................................ 20
1.2.4 Elements of topology on R ......................... 25
1.2.5 The extended real number system ................... 30
1.2.6 The complex number system........................ 31
1.3 Exercises ............................................... 32
1.4 References and comments ................................ 40
2 Vector Spaces and Metric Spaces .......................... 41
2.1 Vector spaces ........................................... 41
2.1.1 Finite-dimensional vector spaces..................... 41
2.1.2 Vector spaces ..................................... 44
2.1.3 Normed spaces.................................... 48
2.1.4 Hilbert spaces..................................... 49
2.1.5 Inequalities ....................................... 51
2.2 Metric spaces ........................................... 57
2.3 Compact spaces ......................................... 65
2.4 Exercises ............................................... 70
2.5 References and comments ................................ 70
3 Sequences and Series ...................................... 73
3.1 Numerical sequences ..................................... 73
3.1.1 Convergent sequences .............................. 73
3.1.2 Subsequences ..................................... 76