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kita

REAL ANALYSIS LECTURE NOTES, TESTS, HANDOUTS, ETC.

1. John Lindsay Orr's Analysis WebNotes [Univ. of Nebraska-Lincoln]

<http://www.math.unl.edu/~webnotes/home/home.htm>

2. Bert G. Wachsmuth's Interactive Real Analysis [Seton Hall Univ.]

<http://www.shu.edu/projects/reals/cont/index.html>

3. Ian Craw's text for MA1002: Advanced Calculus and Analysis

<http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/notes.html>

4. Lee Larson's Real Analysis Lecture Notes [Univ. of Louisville]

<http://www.math.louisville.edu/~lee/RealAnalysis/realanalysis.html>

5. Curtis T. McMullen's notes for Real Analysis and Advanced Real
Analysis [Harvard Univ.]

<http://www.math.harvard.edu/~ctm/past.html>

6. Gianluca Gorni's various handouts and analysis I, II notes
(in Italian) [Univ. di Udine, Italy]

       <http://www.dimi.uniud.it/~gorni/Dispense/index.html#Dispense>
       <http://www.dimi.uniud.it/~gorni/Analisi1/index.html#Analisi>
       <http://www.dimi.uniud.it/~gorni/Analisi2/index.html#Analisi>

7. Caltech's Math 108a,b,c: Classical Analysis

       <http://www.math.caltech.edu/courses/99ma108a.html>
       <http://www.math.caltech.edu/courses/00ma108b.html>
       <http://www.math.caltech.edu/courses/00ma108c.html>

8. Joel Feinstein's handouts and lecture notes in (a) real analysis
and (b) metric and topological spaces [Univ. of Nottingham]

<http://www.maths.nott.ac.uk/personal/jff/G12RAN/>
<http://www.maths.nott.ac.uk/personal/jff/G13MTS/index.html>

9. MA203: Real Analysis (course notes and previous exams), at the
London School of Economics

<http://www.maths.lse.ac.uk/Courses/MA203/index.html>
<http://www.maths.lse.ac.uk/Courses/ma203.html#exams>

       10. MAT 3135; notes and problem solutions in French on undergraduate
           real analysis (509 .pdf file for the notes) and for Lebesgue
           integration (561 K .pdf file for the notes) [Univ. de Montrщal]

<http://www.dms.umontreal.ca/~giroux/analyse_2.htm>
<http://www.dms.umontreal.ca/~giroux/mesure.htm>

11. Dr. Vogel's Gallery of Calculus Pathologies [Texas A & M Univ.]

<http://www.math.tamu.edu/~tom.vogel/gallery/gallery.html>

       12. THE CALCULUS PAGE PROBLEMS LIST by D. A. Kouba [Univ. of Calif.
           at Davis] The problems in the categories "precise epsilon/delta
           definition", "continuity of a function", "Squeeze Principle",
           and "limit definition of the derivative" contain some problems
           (all with solutions) that are sufficiently sophisticated for
           an intermediate level undergraduate real analysis course.

<http://www.math.ucdavis.edu/~kouba/ProblemsList.html>

13. Maria Girardi's tests for Real Analysis (undergraduate),
Analysis I and II (graduate) [Univ. of South Carolina]

<http://www.math.sc.edu/~girardi/w554.html>
<http://www.math.sc.edu/~girardi/w7034.html>

14. Noel Vaillant's Probability Tutorials (Extensive lecture notes
on graduate level real analysis topics.)

<http://www.probability.net/>

       15. Chris Hillman's notes "What is Hausdorff Dimension?" (331 K .ps
           file, 12 page output, for 1995 version; I believe an expanded
           version is being prepared.)

<http://www.math.washington.edu/~hillman/PUB/newhd.ps>

       16. John Shackell's notes for MA 571 Real Analysis and Metric Spaces
           (Univ. of Kent at Canterbury) ["Notes on subsequences" (114 K .ps
           file, 5 page output); "Notes on Riemann integration" (132 K .ps
           file, 12 page output); "Notes on metric spaces" (207 K .ps file,
           16 page output); "Notes on uniform convergence" (146 K .ps file,
           10 page output)]

<http://www.ukc.ac.uk/IMS/maths/people/J.R.Shackell/MA571/ma571.html>

       17. Timothy Gowers "Mathematical discussions contents page". [Topics
           under "Analysis" include: "A dialogue concerning the existence of
           the square root of two"; "The meaning of continuity"; "How to
           solve basic analysis exercises without thinking"; Proving that
           continuous functions on the closed interval [0,1] are bounded";
           "Finding the basic idea of a proof of the fundamental theorem of
           algebra"; "What is the point of the mean value theorem?"; "A tiny
           remark about the Cauchy-Schwarz inequal! ity"]

<http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html>

18. Michael Filaseta's notes for Math 555: Real Analysis II [Univ.
of South Carolina]

<http://www.math.sc.edu/~filaseta/courses/Math555/Math555.html>

       19. Kenneth Kuttler's notes for math 541: real analysis [Brigham
           Young Univ.] (3896 K .ps file, 538 page output) [In addition to
           the standard topics in a graduate real analysis class, these
           notes include chapters titled: "Fourier series", "The Frechet
           derivative", "Change of variables for C^1 maps", "Fourier
           transforms" (includes sections on distributions), "Brouwer
           degree", "Differential forms", "Lipschitz manifolds", and "The
           generalized Riemann integral".]

<http://www.math.byu.edu/~klkuttle/math541notes.ps>

20. Math 105A: Real Analysis [Univ. of California at Santa Cruz]

<http://orca.ucsc.edu/math105a/contents.html>
<http://orca.ucsc.edu/math105a/old_notes.html>

21. Yuri Safarov's notes for Real Analysis CM 321A [King's College
London]

<http://www.mth.kcl.ac.uk/~ysafarov/Lectures/CM321A/>

22. Karen E. Donnelly's notes for M345 Real Analysis [Fall 2000 at
Saint Joseph's College]

<http://www.saintjoe.edu/~karend/m345/>

23. Erhan Cinlar and Robert J. Vanderbei's book "Real Analysis for
Engineers" (483 K .pdf file, 119 page output)

<http://www.princeton.edu/~rvdb/506book/book.pdf>

       24. Vitali Liskevich's [Univ. of Bristol] "Lecture Notes on Measure
           Theory and Integration" (97 K .dvi file, 40 page output) AND
           notes for Analysis - 1 (331 K .dvi file; 1283 K .ps file,
           137 page output)

<http://www.maths.bris.ac.uk/~pure/staff/maval/c98.dvi>
<http://www.maths.bris.ac.uk/~pure/staff/maval/an1.html>

25. Notes for MATH 2610 HIGHER REAL ANALYSIS - Session 1, 2001
at Univ. of New South Wales

<http://www.maths.unsw.edu.au/ForStudents/courses/math2610/notes.html>

       26. James H. Money's "Notes for analysis" ["Definite Integral" (111 K
           .pdf file; "Cauchy Sequences" (126 K .pdf file); "LUB Proof"
           (66 K .pdf file)]

<http://www.math.jmu.edu/~jmoney/analysis/>

       27. Stephen Lich-Tyler's "Who wants to be a mathematical economist?"
           [Univ. of Texas at Austin] (See: 8. Analysis: basics of real
           analysis; 51 K .pdf file, 7 page output)

<http://www.eco.utexas.edu/graduate/Lich-Tyler/math2/>

28. Jim Langley's notes for real analysis [Univ. of Nottingham]
(393 K .pdf file)

<http://www.maths.nott.ac.uk/personal/jff/G12RAN/pdf/JKL.pdf>

       29. Joel H. Shapiro's Lecture Notes [Michigan State Univ.]
           (Includes: "Notes on the dynamics of linear operators"; "The
           Arzela-Ascoli Theorem"; "Notes on Differentiation"; "A Gentle
           Introduction to Composition Operators"; "Nonmeasurable
           sets and paradoxical decompositions")

<http://www.mth.msu.edu/~shapiro/Pubvit/LecNotes.html>

30. Leif Abrahamsson's course material for real analysis
[Uppsala Univ., Sweden]

<http://www.math.uu.se/~leifab/Reellanalys/annat/kursmat00h.html>

       *****************************************************************
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       *****************************************************************

II. SOME USEFUL COLLECTIONS OF LINKS

       1. Dave Rusin's The Mathematical Atlas 26: Real functions and
          Dave Rusin's The Mathematical Atlas 28: Measure and integration.
          See especially "Selected topics at this site" at the bottom of
          each of these web pages.

<http://www.math.niu.edu/~rusin/known-math/index/26-XX.html>
<http://www.math.niu.edu/~rusin/known-math/index/28-XX.html>

2. Bruno Kevius' calculus and analysis links (a huge collection)

<http://www.abc.se/~m9847/matre/calculus.html>

3. Math Forum Internet Mathematics Library: Math Topics:
Analysis: Real Analysis

<http://forum.swarthmore.edu/library/topics/real_a/>

4. The Math Forum: Ask Dr Math: Questions and Answers from Our
Archives: Analysis AND Logic and Set Theory

<http://forum.swarthmore.edu/dr.math/tocs/analysis.college.html>
<http://forum.swarthmore.edu/dr.math/tocs/logic.college.html>

       5. WEB PAGES FOR PH.D. QUALIFYING EXAMS (See the latest version,
          which at present is dated June 23, 2000.) [Virtually all of these
          contain a number of tests (often with solutions) in graduate
          level real analysis.]

<http://forum.swarthmore.edu/epigone/sci.math/howousway>

       6. STUDENT SEMINAR AND SENIOR CAPSTONE REFERENCES [Some of the items
          listed in Section V: "papers, projects, essays by or for students"
          involve real analysis topics.]

<http://forum.swarthmore.edu/epigone/sci.math/gosmangzha/>

7. Chris Hillman and Roland Gunesch's resources for Entropy in
Ergodic Theory and Dynamical Systems

<http://www.math.psu.edu/gunesch/Entropy/dynsys.html>

8. Terence Tao's "Harmonic Analysis Links"

<http://www.math.ucla.edu/~tao/harmonic.html>

9. Online Books and Lecture Notes in Mathematics

       <http://www.gotmath.com/notes.html>
       <http://www.wam.umd.edu/~mcmahonj/notes.html>
       <http://dmoz.org/Science/Math/Publications/Online_Texts/>

       10. IB Higher Level Mathematics: Option 12. Analysis and
           Approximation [A useful list of many topics that arise in
           a honors calculus course or a in lower level real analysis
           course, with links to web pages for more about the topics.]

<http://www.cis.edu.hk/sec/math/Anal&Approx.htm>

11. Michael Botsko's [Saint Vincent College] "A Web Page in Real
Analysis"

<http://facweb.stvincent.edu/academics/mathematics/Ranal/pub.html>

III. SOME OF MY INTERNET POSTS INVOLVING REAL ANALYSIS

*****************************************************************
*****************************************************************

       A. ELEMENTARY TOPICS [Roughly arranged this way: Precalculus topics,
                             differentiation, integration, sequences and
                             series, ODE's, partial differentiation.]

1. Some really complicated surd (i.e. radical) simplifications.

<http://forum.swarthmore.edu/epigone/sci.math/foinulstix/klea641sjsbs@forum.swarthmore.edu>

2. 9 applications of rationalizing the numerator or denominator.
[See the June 13, 2001 post by Dave L. Renfro.]

<http://forum.swarthmore.edu/epigone/ap-calc/mehspyswimp>

3. Why we don't prove a trig. identity as if we were solving an
equation. [Correction given in 2'nd URL.]

<http://forum.swarthmore.edu/epigone/sci.math/chimpleekax/juj2b2dpo63w@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/chimpleekax/5wwhmxx762k7@forum.mathforum.com>

4. Why don't we use negative bases for exponential functions?

       <http://forum.swarthmore.edu/epigone/math-teach/salcrunstou/3w47rj53w6ae@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/math-teach/salcrunstou/arjlxevkkter@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/sci.math/whenspedy/2h2q22wtto7i@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/sci.math/whenspedy/js6r0y0nl0vr@forum.mathforum.com>

5. Remarks about Carl Sagan's novel "Contact", messages in Pi, and
normal numbers.

       <http://forum.swarthmore.edu/epigone/sci.math/glangquingzhal/xkhzfknxw66p@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/math-teach/wingdwimpfrang/xld0w50m889k@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/math-teach/wingdwimpfrang/l8hw4vcqrklz@forum.mathforum.com>

       6. Some remarks about the use of the intermediate value property of
          continuous functions on intervals for solving inequalities and
          8 URL's for worked calculus curve sketching examples. [See the
          May 6, 2001 post by Dave L. Renfro.]

<http://forum.swarthmore.edu/epigone/ap-calc/grithanggix>

7. Some examples where local max/min's of [f(x)]^[g(x)] can be
explicitly found.

<http://forum.swarthmore.edu/epigone/sci.math/whenspedy/ni94df2cep7c@forum.mathforum.com>

       8. Two definitions of an asymptote -- (i) a line the graph approaches
          at infinity; (ii) a line that both the graph and the slope of the
          graph approaches at infinity. [See the Sept. 21, 2000 post by
          Dave L. Renfro.]

<http://forum.swarthmore.edu/epigone/ap-calc/snedikha>

       9. A discussion of L'Hopital's rule and several web pages about the
          continuity and differentiability of x*sin(1/x) and (x^2)*sin(1/x).
          [See the Sept. 24 and 25, 2000 posts by Dave L. Renfro.]

<http://forum.swarthmore.edu/epigone/ap-calc/grisworleld>

       10. A graphical look at a discontinuous derivative and several links
           to other posts of mine that deal with properties of derivatives.
           [See the Dec. 1, 2000 post by Dave L. Renfro.]

<http://forum.swarthmore.edu/epigone/ap-calc/wigufren>

11. Complete details for the integration of (1 + x - x^2)^(1/2).

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/stahglimpchax/040vtkf5jqh3@forum.swarthmore.edu>

12. How to integrate (1 + x^4)^(-1). [Correction given in 2'nd URL.]

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/quumwhendu/zpmaaewv4ha8@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/quumwhendu/1fx0rreqffj7@forum.swarthmore.edu>

       13. Feynman's method of differentiating under the integral sign and
           an excerpt from "Surely You're Joking, Mr. Feynman" where this
           method is mentioned.

<http://forum.swarthmore.edu/epigone/sci.math/wehelcrerm/j0yygwtc7ehu@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/wehelcrerm/6htqjujgxauu@forum.mathforum.com>

14. How to evaluate integrals from -infinity to +infinity of
(polynomial)*exp(-a*x^2 + b*x + c) for a > 0.

<http://forum.swarthmore.edu/epigone/sci.math/swahstimpquer/2svesurt37fw@forum.mathforum.com>

15. Comments and references about series that converge or diverge
very slowly.

<http://forum.swarthmore.edu/epigone/sci.math/pendsnenpun/j51fl2796dss@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/pendsnenpun/8uz9myo1742u@forum.mathforum.com>

16. The Taylor series method for evaluating limits.

<http://forum.swarthmore.edu/epigone/ap_calc/cryphohoi/so22netkq0ij@forum.swarthmore.edu>

17. A survey of a lot of elementary ways to evaluate limits.

<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/3lmtwptgn2vq@forum.swarthmore.edu>

18. Concerning the limit [tan(sin x) - sin(tan x)]/(x^7) as x --> 0.

       <http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/at3or5mb9pjp@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/mnb2imon50dk@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/9euevatpud1x@forum.mathforum.com>

       19. Excerpt from R. P. Agnew's differential equation text about some
           subtleties involved in factoring differential equations. The
           discussion involves an application of Rolle's theorem.

<http://forum.swarthmore.edu/epigone/ap-calc/sixflarten/1js7yqdh0d5i@forum.swarthmore.edu>

20. How to prove 1=2 by partial differentiation.

<http://forum.swarthmore.edu/epigone/sci.math/zheldzermsna/xkpcu7ubdwbl@forum.mathforum.com>

*****************************************************************
*****************************************************************

       B. INTERMEDIATE TOPICS [Roughly arranged this way: logic and set
                               theory, number theory related, metric and
                               topological properties of subsets of R^n,
                               continuity and differentiability behavior
                               of functions, inte! grability issues.]

1. Three proofs that a set with n elements has 2^n subsets.

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/neesnimpyim/catnz4k91efs@forum.mathforum.com>

2. Some web pages having examples of mathematical induction proofs.

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/sparfloxspix/9fo7r86i85xh@forum.swarthmore.edu>

       3. Some REALLY large numbers. [The 4'th URL gives some remarks by
          David Libert (June 14 and 15, 2001) on the Howard ordinal. The
          2'nd group of URL's is a lengthy essay by Robert Munafo.]

       <http://forum.swarthmore.edu/epigone/sci.math/gunsnersteh/8ougi28u0ruv@forum.swarthmore.edu>
       <http://forum.swarthmore.edu/epigone/sci.math/gunsnersteh/b2p89zeylnze@forum.swarthmore.edu>
       <http://forum.swarthmore.edu/epigone/sci.math/gunsnersteh/l8168b3zn68c@forum.swarthmore.edu>
       <http://forum.swarthmore.edu/epigone/sci.math/shingvixtwimp>

       <http://home.earthlink.net/~mrob/pub/math/largenum.html>
       <http://home.earthlink.net/~mrob/pub/math/largenum-2.html>
       <http://home.earthlink.net/~mrob/pub/math/largenum-3.html>
       <http://home.earthlink.net/~mrob/pub/math/largenum-4.html>

4. A list of 16 web pages dealing with elementary aspects of the
cardinality of sets, along with my comments on them.

<http://forum.swarthmore.edu/epigone/sci.math/clerdmandprim/7flq8ianp36g@forum.mathforum.com>

5. Some texts on elementary set theory and my comments on them.

<http://forum.swarthmore.edu/epigone/sci.math/crahseenee/i3406y15vuc8@forum.swarthmore.edu>

6. An essay on how far the transfinite sequence of cardinal numbers
extends, going well past the first ordinal b such that b = Beth_b.

<http://forum.swarthmore.edu/epigone/sci.math/clerdmandprim/h6yjocrbl18v@forum.mathforum.com>

7. Some references for Cohen's CH independence result for someone
just beginning (plus some related internet sites).

<http://forum.swarthmore.edu/epigone/sci.math/zexglysnoo/3uwd7vnlfkj2@forum.swarthmore.edu>

8. Proofs that there are infinitely many primes using relatively
advanced mathematical ideas.

<http://forum.swarthmore.edu/epigone/sci.math/thoxquankee>

9. References for the prime number theorem and the divergence of
the harmonic prime series.

<http://forum.swarthmore.edu/epigone/sci.math/ryblilfral/37E88457.F56AF600@gateway.net>
<http://forum.swarthmore.edu/epigone/ap-calc/stelthenddwel/s3sn0phwayms@forum.mathforum.com>

       10. Extensive list of references (print and internet) for proofs
           that (1) e is irrational, (2) Pi is irrational, (3) e is
           transcendental, and (4) Pi is transcendental.

<http://forum.swarthmore.edu/epigone/sci.math/zhingpheldkoi/zmek2cmd0gdh@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/zhingpheldkoi/5jitzz6z85ba@forum.mathforum.com>

11. Proof that a trig. function of a rational number of degrees is
an algebraic number, along with several literature references.

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/vogoubril/27t25cke83lo@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/vogoubril/cq3ssizdywsg@forum.swarthmore.edu>

       12. Brief discussion of whether SIN(1 deg) is (a) a constructible
           number (in the ruler and compass sense), (b) an explicit
           algebraic number, and/or (c) an algebraic number. Explicit
           expressions (using radicals and rational numbers) for the exact
           values for SIN(3 deg) and SIN(180/17 deg) are given.

<http://forum.swarthmore.edu/epigone/math-teach/sheltolla/ikd4eb70o126@forum.mathforum.com>

13. Remarks about explicit algebraic numbers and algebraic numbers.

       <http://forum.swarthmore.edu/epigone/sci.math/poxcrimpvo/2p0zdwozs0zn@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/sci.math.symbolic/playzerdblar/pqd9rd0x3wui@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/sci.math.symbolic/playzerdblar/dw03ob591pfk@forum.mathforum.com>
       <http://forum.swarthmore.edu/epigone/sci.math.symbolic/playzerdblar/k! f2fw9x8n7f7@forum.mathforum.com>

14. Two proofs that any closed set of real numbers is equal to the
set of cluster points of some sequence.

<http://forum.swarthmore.edu/epigone/sci.math/zhimpprudwel/t09kobzpef45@forum.mathforum.com>

15. Some fractal dimension, and other, results concerning
Cantor sets constructed by a dyadic process.

<http://forum.swarthmore.edu/epigone/sci.math/thaxprenvil/97lr7ennygt5@forum.swarthmore.edu>

       16. Some references and historical remarks about a result that
           Cantor proved in 1882 -- If D is a countable subset of R^n,
           then R^n - D is pathwise connected.

<http://forum.swarthmore.edu/epigone/sci.math/blexspehwhin/q4du0d1qe4b6@forum.mathforum.com>

       17. Discussion about and web page references to continuity
           matters--especially of the ruler function. A proof is given
           that the ruler function is continuous at each irrational point
           and discontinuous at each rational point.

<http://forum.swarthmore.edu/epigone/ap-calc/daxkookay/382D63E9.D43F2111@gateway.net>

18. A discussion of various notions of "increasing at a point" vs.
"increasing on an interval" vs. "having a positive derivative".

<http://forum.swarthmore.edu/epigone/ap-calc/rulmaybrox/ruq95jibfnsw@forum.mathforum.com>

       19. Geometric discussion of locally linear and a generalization due
           to Bouligand that applies to any subset of the plane (the
           contingent of a set). [See Nov. 27, 2000 post by Dave L. Renfro.]

<http://forum.swarthmore.edu/epigone/ap_calc/grandzaygren>