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Pure Math

  1. Nov 23, 2009 #1
    Are there any must have pure math books for mathematicians, or just math in general? Christmas is around the corner so I'm lining up some books. I already have a number theory book by George Edwards and a Set Theory/Logic book from Robert Stoll. I'm going to be getting The Principles Of Mathematics and How to Prove It: A Structured Approach.

    Thank you
     
  2. jcsd
  3. Nov 23, 2009 #2
    There are a wide variety of topics and subjects in mathematics (as you surely already know), so it's rather hard to pick just a few. My (mixed special-and-general topic) library currently includes the following subjects and their respective books. I've listed only the ones that I consider to be very useful and interesting.

    Algebra

    Algebra MacLane (My personal favorite)
    Algebra Hungerford
    Algebra Lang
    Finite-Dimensional Vector Spaces Halmos

    Analysis

    Principles of Mathematical Analysis Rudin
    Real Analysis Royden
    Visual Complex Analysis Needham
    Functional Analysis Rudin
    Real and Complex Analysis Rudin (This is perhaps the best analysis book out there)
    Measure Theory Halmos
    Introduction to Analysis Rosenschlict (Dover)


    Topology

    Topology Hocking and Young (Dover)
    Topology Munkres
    Topology Dugundji (Fabulous, if you can find it for a decent price, buy it!)
    Algebraic Topology Hatcher

    Set Theory and Classical Logic

    First-Order Mathematical Logic Margaris (Dover)
    Axiomatic Set Theory Suppes (Dover)
    Naive Set Theory Halmos
    Set Theory Jech (unbelievable monograph)
    Logic: Techniques of Formal Reasoning Kalish (My favorite introduction to logic, it is dense and fun)

    Category Theory and Nonclassical Logic- I have a bias with this subject in particular; Categories and Nonclassical Logics fascinate me to no end.

    Categories for the Working Mathematician MacLane (One of my absolute favorite math books)
    Topos Theory Johnstone
    Topoi: The Categorial Analysis of Logic Goldblatt (Dover)
    First Order Categorical Logic Makkai
    An Introduction to Higher-Order Categorical Logic Lambek (A fantastic book)

    Have fun =P
     
  4. Nov 23, 2009 #3
    woah one great collection you got there :)
    thanks for sharing.
     
  5. Nov 24, 2009 #4
    Reedeegi's list is pretty comprehensive, but I'll offer two recommendations:

    1. Kolmogorov and Fomin's Functional Analysis is great if you know some basic analysis. The chapter on metric spaces (the chapter after the conventional first chapter on set theory in many undergraduate texts) is superb, as the authors don't hesitate to present some of the most important metrics in analysis with great detail. A lot of the basic topology needed for analysis is also introduced, and the material on normed linear spaces looks good (I'm on this part now). The last couple of chapters are on measure and integration, so the text covers both functional analysis and modern real analysis.

    2. Counterexamples in Topology. This text helped me learn the general topology needed for my current analysis course. The very short and condensed introduction covers the basic facts on general topology, and quickly moves onto the various examples of topological spaces. While a lot of important theorems such as those due to Tychonoff or Urysohn aren't proven here, you'll learn a lot about topological properties by studying the examples (and counterexamples are pretty handy). Ideally, you would want to supplement a general topology text with this one, but I find that the introduction along with wikipedia suffices to understand most of the examples in the text.
     
  6. Nov 24, 2009 #5
    After checking Dover's website, I also noticed that they have quite a few "introduction to" books, some of which I've heard are quite good:

    Elements of Abstract Algebra Clark
    Abstract Lie Algebras White (If you want to do any mathematically-rigorous physics, this book will help.)
    Introduction to Topology Mendelson (Point-set topology)
    Introduction to Topology Gamelin (Topology for analysis)

    Also, one book I forgot to mention (also published by Dover, coincedentally) is Abstract and Concrete Categories: The Joy of Cats by Adamek et al. My only gripe with it is that it uses set theory as a foundation for categories, but it is nevertheless a useful title.
     
  7. Nov 25, 2009 #6

    jbunniii

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    I highly recommend Courant and Robbins, "What Is Mathematics?"
     
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