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Beginning real analysis

  1. Mar 17, 2014 #1
    I'd like to start learning at home real analysis.
    Now, in order to start there was an older book Techniques of mathematical analysis by Tanter which looks good as preparation.
    I also saw that another user SanjeevGupta studied the same one, and found it good.
    I'd like to see some comments on this. If someone knows how I can reach SanjeevGupta on this forum
    please tell me, I'm new here.
    I don't really like to restart calculus. Apostol could be an alternative but is somewhat more difficult.
    The Tanter book offers quite some different interesting subjects to start with, including Diff. Geometry intro at the end. Books like advanced calculus are different in scope.

    As I'm new, Latex is also new for me. How can I learn this here ?

    The problems I have, these I would like to address here ...

    Thank you
  2. jcsd
  3. Mar 17, 2014 #2
    You could send him a private message by clicking on his name you saw the thread.

    Also I just started learning analysis too. I found these lectures.
    http://www.uccs.edu/math/student-resources/video-course-archive.html [Broken]

    There's intro level then the higher level that goes into measure theory and such.
    Last edited by a moderator: May 6, 2017
  4. Mar 17, 2014 #3
    I will say that Bartle's real analysis is a very good book or even advanced calculus by Fitzpatrick. Try those out as well.
  5. Mar 18, 2014 #4
    Were these just video lectures without studymaterial ?
    Were the videos readable (some use blackboard on youtube and totally unreadable)?
    If there is no studymaterial ? What about exercises ? How to capture the theory on your mac or pc, to have at least a copy.
    Just videolectures is for math clearly insufficiant.

    I see that there are or lecture notes scribbles, or video problems, or you can email the prof on Modern analysis 1
    for his course. That's it I guess.
    Problem: Videos won't start on my mac . I've got QTP 10.0.

    Which courses did you take ?
    Also on proof reading and writing studystuff is needed. I ordered a book on that.
    I know, lot of questions, but I'm trying this now for quite a number of time, also being not good by health, and support material to read on a tablet is indispensable.
    I did maths 1 year at univ 30 years ago, and without proper preparation on proofs the gap between calculating calculus in high school and the huge abstract syllabus on analysis was too big for me. I studied in Ghent Belgium. Nowadays they have lowered the difficulty degree, but then,
    horendous (misspelled ?).

    Thanks very much.

    Thanks for the response. I will certainly have a look.

    Videos don't start on my mac in QTP 10.0.
    There are indeed lecture notes, videos with problems, some courses, and I've sent the prof on Modern Analysis 1 a mail for his handbook.
    Are there more resourses ?
    Last edited by a moderator: May 6, 2017
  6. Mar 18, 2014 #5
    I have download copies of those as well.
    Are you keen on :
    Advanced Calculus (A Transition to Analysis) - Joseph B. Dence and Thomas P. Dance

    Having to bridge 30 years, which approach would you take ?
  7. Mar 18, 2014 #6
    In my experience, LaTeX is definitely a program you need to learn hands-on. Start an assignment/proof in LaTeX, and look up symbols/code as you go. There's a website called detexify that's sometimes helpful, and I also have used the wikibook on LaTeX a lot.
  8. Mar 18, 2014 #7
    I just use my own book and work problems out of that. I'm using Foundations Modern Analysis by Friedman, although it's not very introductory level.

    The videos play on my Mac in Chrome. Quicktime 10. Although I just download them and use VLC by looking at the page source in Chrome.

    These are not the Harvey Mudd lectures from YouTube where the board is hard to read. It is clear.

    The Modern Analysis videos sort of start by trying to show how to write proofs. I read a book called "How to Prove It" by Velleman to learn proof writing. I was a physics major, so I've never actually taken a proof based course.
  9. Mar 18, 2014 #8
    Can a good analysis curriculum begin without calculus, meaning starting by a book on proofs which I ordered, and then step into real analysis. I think yes. From a prof in Brussels, I could follow her 2nd years analysis course If I first had mastered the basic material on sets, proofs...
  10. Mar 18, 2014 #9
    Prof at the ULB or VUB?
  11. Mar 18, 2014 #10


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    If you have learned (nonrigorous) calculus then you can start with introductory real analysis. Rigorous calculus books such as Spivak and Apostol are essentially introductions to analysis that don't assume you already know any calculus. Just don't try to start with a book that assumes you have already been exposed to rigorous calculus (epsilon-delta proofs). Bartle would be a good choice. Rudin would not.
  12. Mar 18, 2014 #11
    VUB Colebunders
  13. Mar 18, 2014 #12
    And you could actually follow her second year course? It requires you to be familiar with metric spaces, Banach spaces, uniform norms, epsilon-delta arguments, etc.

    The first section starts off with a study of uniform convergence and of sequences of functions and is immediately filled with a lot of epsilon arguments. The second section is a study of power series in context of Banach spaces. And then she goes to do power series rigorously, culminating in the wonderful identity theorem for analytic functions.

    I find it weird you say you only need proofs and sets to tackle the course. Either that, or she made her course a lot easier.
  14. Mar 18, 2014 #13
    Real numbers and set theory Mark Sioen. This demands Analysis I as prerequisite:

    Elementary set theory with attention to quotient, products and factorization of functions
    Cauchy construction of R
    Algebraic properties of R
    Order properties of R
    Q as subfield of R
    Basics of metric and normed spaces
    Completeness of R
    Unicity up to isomorphism
    Decimal representations
    The completed real line as complete lattice
    Lipsup and liminf of bounded sequences in R
    Size of sets, countability
    Size of N, Z, Q, R

    Analysis 2 by Mark Sioen:

    Compactness and connectedness of metric spaces, in particular for R and C
    Pointwise and uniform convergence of sequences of functions. Spaces of functions equipped with sup-norm
    Completeness and applications on the fix point theorem
    Dini's theorem. Differentiating and integrating limit functions
    Series: convergence and absolute convergence; series of real and complex numbers, Abels and Dirichlets criterium, operations on series, absolute summability
    Sequences of functions, Weierstrass M test, Abels and Dirichlet's criterium, Taylor series, power series in C. Analytic functions and analytic continuation

    I was allowed to take this second course, but things change...
    Last edited by a moderator: Mar 18, 2014
  15. Mar 18, 2014 #14
    I'm going to have to ask you to keep posting in english (forum rules).

    Anyway, there is no way you can handle Analysis II without knowing metric spaces rigorously and without being able to perform epsilon-like proofs. Proofs with Cauchy sequences and the like should be something you can do from the top of your head.

    The class of "real numbers and elementary set theory" is something you can probably take now, but it's heavy on proofs. Also, the Cauchy construction of the reals will make no sense what-so-ever if you never encountered cauchy sequences before.
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