Problems and Solutions for Undergraduate Analysis Rami Shakarchi, Serge Lang

384 Pages·1998·32.19 MB·English

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Rami Shal<archi
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PROBLEMS AND SoLlmONS FOR
UNDERGRADUATE ANALYSIS

Rami Shakarchi
PROBLEMS AND
SOLUTIONS FOR
UNDERGRADUATE
ANALYSIS
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Springer
Rami Shakardli
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Princdoo Uni~tnily
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Preface
The present volumecontainsaU theexercisesand theirsolutionslOr Lang's
second edition of Undergraduate Analysis. The wide variety of exerdllell,
which range from cornputatiolUl1 to mote conOOl1uallUldwhichareofvary
ing difficulty, cover the folJowing subjects and more: real numbers. (jmltll,
continuous functioos, diJferentiation and elementary illtegration, nonoed
vect.oT spaees, eornpa.ctncsa, series, IntegraUon In one \'lI.riabIe, ImJX'OPCl"
integrals, convolutions, Fourier series and the Fourier Inkgrsl, funcUons
in n·space, derlvatlVfJa In vector spaces, the IIM!l'8e and Impiclt mapping
theorem, ordinary differentialequations, multiple Integrsls, and dilferenUal
Jonns. My ob!ectiw Is to offer those learning and teaching analyllis at the
uode:rgradua.te level II Ia.q:e number ofoompl<lted exercieesand I hope that
this book, which c«etains over600exercisesooverln,g the topics ~ioned
above, will acbieVfl my go&!.
The exercillell are au integral pe.rt of Lang'lI book and I encou.rage the
reader to work through all of them. In some cases, the problemll in the
beginningchaptelllare used in laterones, lorexample, in ChapterIV when
one CXJUtructs-bump functions. which are used to 8l1l00th out singulari
ties. and prove that the space of funetiom Is dense in the space of ~
Iated m&pll. The IlUmberlng of the JYOblmul is as lol1OWll. Exercise IX.ti.7
.in"dic"ate"s Ex"erc-ise 7, §5, ofChapter IX.
w
I lUll grateful Serge Lang for his help and ent.husl88m In this project,
8Il well as for teaching me mathemat.ic5 (and IDllclJ Dl()£(l) with 110 nmch
generosity aOO patleoce.
My ramUy'8 8IIppOI't proved illVllluabJe. I thank my parentsand brother
roc a1wa,ys bemg there when needed.
I thank the faculty at the Yale Mathematics Departmem. fur their en
couragemems Md help In the past four years, as well as fot the time they
have 6pCIIt te&eb1n8 me mathematice.
Fmally, I am indebted to Stevm Jool Miller lot his of humoc and
IIeI1lle
~ support and IIBSistance when we do mathematics klgether.
YaIe,l997