Real and Complex Analysis Volume 1

645 Pages·2018·4.27 MB·English

by  Rajnikant Sinha

Rajnikant Sinha
Real and
Complex
Analysis
Volume 1
Real and Complex Analysis
Rajnikant Sinha
Real and Complex Analysis
Volume 1
123
Rajnikant Sinha
Varanasi, Uttar Pradesh, India
ISBN978-981-13-0937-3 ISBN978-981-13-0938-0 (eBook)
https://doi.org/10.1007/978-981-13-0938-0
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Preface
The book is an introduction to real and complex analysis that will be useful to
undergraduate students of mathematics and engineering. It is designed to equip the
readerwithtools thatwill help themtounderstandthe conceptsofrealanalysis and
complex analysis. In addition, it contains the essential topics of analysis that are
neededforthestudyoffunctionalanalysis.Itsguidingprincipleistohelpdevelopthe
necessary concepts rigorously with enough detail and with the minimum prerequi-
sites.Further,Ihaveendeavoredtomakethisbookbothaccessibleandreadable.This
book contains complete solutions to almost all the problems discussed within. This
willbebeneficialtoreadersonlyifusedcorrectly:readersareencouragedtolookat
the solution to a problem only after trying to solve the problem.
Certainly,attimes,thereadermayfindtheproofsexcruciatinglydetailed,butit
is better to be detailed than concise. Furthermore, omitting the detailed calculation
can sometimes be perplexing for beginners. I have tried to make it a readable text
that caters to a broad audience. This approach should certainly benefit beginners
who have not yet tussled with the subject in a serious way.
This book contains several useful theorems and their proofs in the realm of real
and complex analysis. Most of these theorems are the works of some the great
mathematicians of the 19th and 20th centuries. In alphabetical order, some of these
include: Arzela, Ascoli, Baire, Banach, Carathéodory, Cauchy, Dirichlet, Egoroff,
Fatou, Fourier, Fubini, Hadamard, Jordan, Lebesgue, Liouville, Minkowski,
Mittag-Leffler, Morera, Nikodym, Ostrowski, Parseval, Picard, Plancherel, Poisson,
Radon,Riemann,Riesz,Runge,Schwarz,Taylor,Tietze,Urysohn,Weierstrass,and
Young. I have spent several years providing their proofs in unprecedented detail.
There are plenty of superb texts on real and complex analysis, but there is a
dearth of books that blend real analysis with complex analysis. Libraries already
contain several excellent reference books on real and complex analysis, which
interested students can consult for a deeper understanding. It was notmy intention
to replace such books. This book is written under the assumption that students
alreadyknowthefundamentalsofadvancedcalculus.Theproofsofvariousnamed
v
vi Preface
theorems should be considered to be at the core of the book by any reader who is
serious about learning the subject.
The book is divided into two volumes. Volume 1 contains three chapters:
Lebesgueintegration,Lp-spacesandFouriertransforms.InChap.1,webeginwith
the definition of an exponential function and prove that it maps the set of all
complex numbers onto the set of all nonzero complex numbers. After that, we
develop the Lebesgue theory of abstract integration of complex-valued functions.
Next, we prove the Riesz representation theorem in enough detail and use it to
answerthe question: is every set of n-tuples of real numbersLebesgue measurable
in Rn? The theme of Chap. 2 is Lp-spaces. First of all, we introduce convex
functions and then prove the Riesz–Fischer theorem. In the end, we derive some
properties of Banach algebra. We have shown that Lp-spaces is an example of a
Banach space. In Chap. 3, we introduce total variation, and prove the Radon–
Nikodym theorem. Fubini theorem and the change-of-variable theorem are also
proved in this chapter. Finally, we discuss the Plancherel theorem on Fourier
transforms.
I am particularly indebted to Walter Rudin and Paul Richard Halmos for their
letters discussing academic questions. By great good fortune, some colleagues of
minewereabletojoininwiththisenterpriseafewyearsago,someofwhomhave
provideda meticulous reading of themanuscript from a user’sviewpoint. Iextend
my great thanks to all of them for their expert services.
Whilestudyingthisbook,Ihopethatreaderswillexperiencethethrillofcreative
effort and the joy of achievement.
Varanasi, India Rajnikant Sinha
Contents
1 Lebesgue Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Integration of Positive Functions . . . . . . . . . . . . . . . . . . . . . . . . 51
1.4 Integration of Complex-Valued Functions . . . . . . . . . . . . . . . . . 85
1.5 Sets of Measure Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
1.6 Preliminaries to Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1.7 Preliminaries to Riesz Representation Theorem. . . . . . . . . . . . . . 139
1.8 Riesz Representation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 167
1.9 Borel Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
1.10 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
1.11 Existence of Non-Lebesgue Measurable Sets . . . . . . . . . . . . . . . 216
2 Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
2.1 Convex Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
2.2 The Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
2.3 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
2.4 Orthogonal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
2.5 Riesz-Fischer Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
2.6 Baire’s Category Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
2.7 Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
2.8 Banach Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
3.1 Total Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
3.2 Radon–Nikodym Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
3.3 Bounded Linear Functionals on Lp. . . . . . . . . . . . . . . . . . . . . . . 431
3.4 Lebesgue Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
3.5 Metric Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
3.6 Vitali–Caratheodory Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 510
vii
viii Contents
3.7 Change-of-Variables Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 535
3.8 Fubini Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
3.9 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
3.10 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
3.11 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
3.12 Inversion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
3.13 Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 637
About the Author
RajnikantSinha isFormerProfessorofMathematicsatMagadhUniversity,Bodh
Gaya,India.Asapassionatemathematician,hehaspublishednumerousinteresting
research findings in international journals and books, including Smooth Manifolds
(Springer)andthecontributedbookSolutionstoWeatherburn’sElementaryVector
Analysis. His research focuses on topological vector spaces, differential geometry
and manifolds.
ix
Chapter 1
Lebesgue Integration
InadequaciesoftheRiemannintegration,anddifficultiesinhandlinglimitprocesses
in it were largely overcome with the advent of abstract integration. Next, upon
applying a remarkable result—the Riesz representation theorem—Lebesgue mea-
sure in Euclidean space is introduced. Just like the real number system is a com-
pletion,inacertainsense,oftherationalnumbersystem,Lebesgueintegrationisa
completion, in a certain sense, of Riemann integration. These phenomena will be
vividly demonstrated in this somewhat long chapter. We begin with the definition
ofexponentialfunction,andprovethatitmapsthesetofallcomplexnumbersonto
thesetofallnonzerocomplexnumbers.Afterthat,wedeveloptheLebesguetheory
of abstract integration of complex-valued functions. Next we prove the Riesz
representationtheoreminsufficientdetailanduseittoanswerthequestion:Isevery
set of n-tuples of real numbers is Lebesgue measurable in Rn?
1.1 Exponential Function
Since exponential function will occur quite frequently in later chapters, it seems
prudenttolayagoodfoundationforthisattheearliestopportunity.Thissectionis
devoted to this end.
Note 1.1 For every complex number z,
(cid:1) (cid:1)
lim(cid:1)(cid:1)(cid:1)ðnznþþ11Þ!(cid:1)(cid:1)(cid:1)¼ lim jzj ¼jzj(cid:3)lim 1 (cid:4)¼jzj(cid:2)0¼0\1;
n!1(cid:1) znn! (cid:1) n!1nþ1 n!1nþ1
so, by the ratio test of convergence, the series 1þzþ z2 þ z3 þ (cid:2)(cid:2)(cid:2) is absolutely
2! 3!
convergent for every complex number z.
©SpringerNatureSingaporePteLtd.2018 1
R.Sinha,RealandComplexAnalysis,
https://doi.org/10.1007/978-981-13-0938-0_1