Analysis I
     110.405,  ThFr 11-12:15,  Room 225 NEB
Professor:  Christina Sormani
                          Kriegar 220,  516-6637                Office Hours: Mondays
                                    10:30-11:30, 3-4, 5-6

Texts:  Elementary Classical Analysis (required),  
                            by Marsden and Hoffman, 2nd Edition; Freeman Press
                            Proofs are at the end of chapters and must be read.
             The Way of Analysis (recommended),
                           by Strichartz; Jones and Bartlett Press
                           Useful for learning how to prove.

Notes:  Supplementary Handouts will be used especially in the
                        beginning of the course to teach proving techniques.

Prerequisites:   Multivariable Calculus and Linear Algebra are
                required. This course is significantly easier to handle if you
                have had experience proving theorems.  It is recommended that
                you review proofs you have studied in the past including epsilon
                delta proofs from Calculus.

Grading Policy:

       Homework, 40%:   The problems in this course are difficult and
              cannot be completely tested on exams.  Thus the homework is
              a significant portion of your grade.  Sometimes you will be
              required to resubmit an assignment to improve your grade.
              No late assignments will be accepted without a doctor's
              note.    Assignments are due at the beginning of class if due
             Thursday and at 5pm if due Tuesday.  You may consult with
             classmates but be sure to do most of the work yourself.
             Starting after week 4, homework due on Thursdays will be short
             assignments on old material which will be graded in their entirety.
             Homeworks due on Tuesdays will be as before.

     Exam I,  15%  This exam was scheduled after the fourth week.
             It consisted of four problems, the first three on a Monday evening
             with no time limit (~2 hours) and the fourth on a Thursday with a 20 min
             time limit.  Problem 1 was a choice of one of two proofs by
             contradiction (20%).  Problem 2 was a choice of one of two limit proofs (20%).
             Problem 3 was a choice of two four part problems one on intersections
             the other on equivalence classes (40%).  Problem 4 is a harder proof by
             contradiction (20%) see the email.

   Exam II, 15%:  This exam will cover the topology of metric spaces:
              continuity, open sets, closed sets, boundaries, compactness,
              connectedness etc.  It will be given one week after this material is
      Final 30%:    This exam will be given during the finals week.

Syllabus:   (This syllabus will be updated as the course progresses)
        Week 1:  Introduction to  Analysis, What is Proof?, Handout,
                           Contradictions, Mappings, Equivalence Classes, Handout
                   HW:  Handout 1-3, all problems, due Tuesday Sept 8
                            Handout  4-5, all problems, due Thursday Sept 10
                            Our new classroom is 225 NEB behind Shaffer Hall.
                           If you did not get email about this, send me an email!
        Week 2:   Denumerability and  the Rationals, Handout, 1.1
                            Sequences in the Rationals, Handout, 1.2
                   HW:  Handout 6-7, all problems, due Tuesday Sept 15
                            Handout 8, all problems, 1.1/5 due Thursday Sept 17

        Week 3:   Construction of the Real Line and Completeness
                             Unique Decimal Expansions, [0,1] is not denumerable
                   HW:   Handout 9-10, due Tuesday Sept 22
                             1.2/1, 2, 4, 5   due Thursday Sept 24

        Week 4:   Euclidean space, Cauchy Schwartz, Complex , 1.6,1.8
                             Metric Spaces 1.7,
                   HW:   Due Thursday Oct 1 but good review for the exam:
                             Handout 11 (on my door), all problems
                             (Read all of1.4) 1.4/ 1, 4, 5 (use negation for 5)
                             (Read 1.8 only pp70-72), verify that the complex plane
                              is a field, 1.8/1, 5
                             After the exam, read 1.7

        Week 5:  Exam I Monday Sept 29 6-8 pm or 7-9pm,
                                        Krieger hall Room 300, Problems 1-3
                           Thursday (20 min in class Exam I Problem 4)
                                     If you did not get the email, send a note.
                            Open sets and Interiors,  2.1
                            Interiors and Closed Sets,  2.2-2.3
                    HW due Tue:  2.1/ 1, 3, 4 (draw B if C=[0,1]x{0}) , 2.2/3, 4,  2.3/ 4
                    HW due Th :   1.2/ 4 give explicit counter example or prove with explicit N
                            depending on epsilon, p97/ 3a,b note that ]0,1[=(0,1), p97/6
        Week 6:   Closure, Boundaries, infs and sups 2.4-2.6, 1.3
                             Sequences, Cluster Points, limsups,  2.7-2.8, 1.5
                    HW due Wed: (Will grade *'ed only) 2.7/2*,3*, 2.4/1,2*,3, 6*,
                              2.5/1, 2*, 1.3/2*,  97/1
                    HW due Friday:  2.2/3, 2.3/1,2*,6*,1.4/3, 98/10, 99/15  
        Week 7:    Sequencial Compactness 3.1.1
                             Compact Sets and Bolzano Weierstrass 3.1
                     HW due Tuesday: 1.5/ 1,2*,4*(prove or provide a counterexample)
                             2.8/2*,3,4*, 3.1/1,2*,4*,5
                     HW due Thursday:    p143-5/1,2*(no proof), 4*, 7, 8*,9a, 13, 15a*,18*
       Week 8:      Bolzano Weierstrass Continued
                               Euclidean Space and Heine Borel 3.2 (Handout),
                               Hilbert Space (Handout)
                     HW due Wednesday: 3.2/1 , 4*, Heine-Borel Handout/1*, 3*,4*,
                               Hilbert Space Handout/ 1*, 2*, 4*,5, 6*, 7*
                               Show that if  Vi  is a sequence in Hilbert space that
                               converges to  V,  then each component converges
                               to the appropriate component of V.   Give an example
                               showing that even if each component converges, the
                               sequence itself may not converge.

                  Extra Credit from Heine Borel Handout due Friday
                               Hilbert Space/ 3 is also extra credit but no deadline yet.

       Week 9:    Continuity 4.1 (not Thm 4.1.4 iii and iv),
                             Path connected and Connected Sets 3.4-5
                   HW due Tuesday:   4.1/1*, 5, p231/1, problems 1*, 2* and 3* below
                              3.4/1(no proofs),  3.5/1,2*

                         1) Let  g: A to R where A=R\{0}.  Let g(x)= (cos(x)-1)/x.
                              Prove that       lim     g(x)  = 0 using the defn.
                                                    x to 0
                        2) Let F:R to R^2 be defined as F(t)=(x(t), y(t)) where x and y are functions
                             from R to R.  Prove that F is continuous at s iff x and y are continuous at s.
                         3)  Let g: (-pi/2, pi/2) to R be defined as g(x)=tan(x).  a) Prove that
                              g is continuous at any point in (-pi/2, pi/2).   b) prove that pi/2 is
                              an accumulation point of  (-pi/2, pi/2).  c) prove that there does not
                              exist L such that            lim          g(x)    =   L      .
                                                                 x to pi/2
                   HW due Thursday:   p172-6/1, 4*, 5b*,  12*, 14*, 17, 18*,  33, 37,                               

        Week 10: Continuous Images and Preimages of sets 4.1-4.2
                             Intermediate Value Theorem 4.4-4.5 (purely Real)
                    Exam II Monday Nov 9:  All material weeks 4-9 and 4.2.
                             2 hours (come between 6-9pm)  Maryland Hall 114.
                    No Homework this week,

      Week 11:  Uniform Continuity 4.6, Differentiation of functions on the Real line 4.7
                              Integration, Fundamental Theorem of Calculus, 4.8, 8.1
                       HW due Wednesday:  Differentiation: 4.6/1, 2*, 3, 6*, 4.7/2*, and 1* below
                              Integration: p234-6/22*, 44*, 45    8.1/2*,3, 6*  and 2* below
                              Continuity Review: 3* below.
                              1)  Let f(x) and g(x) be differentiable on R, f(y)=g(y)=0 and g'(y) not 0.
                                   Use the mean value theorem and definition of limit to prove
                                   L'hopital's Rule:   limit as x approaches y of f(x)/g(x)  = f'(y)/g'(y).
                              2)  Let F(x)=0 when x is irrational and F(x)=1 when x is rational.
                                    Prove that F is not Riemann Integrable (that is, show that
                                    inf upper sums > sup lower sums).   Hint:  Remember that
                                    between any pair of real numbers, x<y, there is a rational
                                    number q and an irrational r between them.

                              3)  Let  f: R to R be continuous everywhere.  The support of
                                    f is the set A = Cl({ x : f(x) does not equal 0}).  Prove that
                                    if f has compact support then f is bounded.

       Week 12:  The Space of Continuous Functions, Uniform Convergence  5.1, 5.5
                             Integration and differentiation of limits, Function Spaces
                                  (need only know: 5.2.1, 5.3.1, 5.3.3, 5.3.5, 5.3.6,  from 5.2-3).
                     HW due Tuesday Nov 24:   5.1/1, 2*,3, 5.2/1,  5.3/1*, 2*, 3*
                              Function Spaces Handout (use defn of support as above)/ 1*, 2**, 3*, 4*,5*
                     HW due Friday Dec 4:  p174-6/ 21*, 39*,
                               p234-6/ 26*, 38*, 39, 42* , 45,  p316/ 2*, 10 (only sequences from 2)*
                               Review Exam II including problems you didn't choose.

      Week 13:   Volume, Measure Zero and Denumerability 8.2
                   No more HW due.    Please fill out the Teaching Evaluation before the final.
                   Review Problems:  5.1/4, p231-4/1a, 3, 7, 11a (just do f is cont implies graph is path conn),
                             15, 19, 21, 29, 33  p316-318/ 1, 5, 13, 19 (show the sum conv unif on [-r,r] for any r),
                              29ac, 33, 37,  ...

     Final:  Friday December 11, Shaffer Hall Room 1, 2-5pm
                   There will be five parts:
                           I. Short Answers (these will be longer than the ones in the second
                               exam but not as many) negating statements, verifying something
                                is a metric, find limsup/inf with no justification, identify compact
                                open, closed and connected sets, and boundaries in Euclidean space.
                           II. (Chose between two explicit sequences of functions),
                               Prove or prove not uniformly conv, prove if the derivatives
                               conv and/or the integrals. (be sure to know theorems of week 12
                               and techniques I went over last Thursday)
                          III. (chose one of two) Proofs regarding uniform continuity,
                                Riemann Integrability and differentiability. (week 11)
                                (Know these definitions)
                           IV. (chose one of two) Proofs about connectedness and
                                  continuity in metric spaces (week 9-10)
                           V. (chose one of two) Proofs about the metric space properties of sets in
                                C([0,1]) (these will include new concepts)  Be sure to know the
                                definitions of closed, open, compact, boundary, connected, closure,
                                interior so that you can apply them here as well as in I.  Write down
                                the definitions before you apply them.  (weeks 5-7,12)