UNIVERSITY OF WAIKATO

Department of Mathematics math501-09B

Metric Spaces
Today we will continue with revision. The date for the final test will be set and office hours outlined. An example test is linked here. The list of topics is given directly below.

  1. All definitions and theorem statements
  2. min{d,M} for M>0 is a metric equivalent to d
  3. a convergent sequence in (X,d) is Cauchy. In R, Cauchy implies convergent
  4. in R, sufficient conditions via Taylor for a local minimum, point of inflection
  5. Taylor expansions derived

  6. characterization of Riemann integrable (RI) function on [a,b]
  7. if f is RI and c>0 so is c.f and int c.f = c int f
  8. f<= g and both RI implies int f <= int g
  9. f:[a,b]->R continous implies uniformly continuous
  10. continuous implies RI
  11. integral as a continuous or differentiable function of the upper limit
  12. uniform limit of continuous/RI functions is continuous/RI

  13. closure interior and boundary in a topological space
  14. metrizable topology is Hausdorff
  15. f continuous iff the inverse image of every closed set is closed
  16. closed subsets of a compact space is compact
  17. in a Hausdorff space compact subsets are closed
  18. continuous image of a compact/connected set is compact/connected
  19. in R^n compact subsets are closed and bounded
  20. if A is connected so is its closure
  21. (a,b) is connected
  22. second countable implies separable
  23. in a metrizable space separable implies second countable
  24. metrizable spaces are normal
  25. compact Hausdorff spaces are normal
  26. contraction mapping theorem

  27. derivative on R^n is unique when it exists
  28. differentiable implies continuous
  29. matrix representation for derivatives f:R^n -> R
  30. derivative of a linear map
  31. all norms are equivalent on R^n

  32. C[a,b] as a normed space with inequivalent norms
  33. algebras of functions closed under max and min
  34. Cauchy-Schwartz inequality
  35. projection theorem in a Hilbert space
  36. projection is linear
  37. properties of complete orthonormal sets in Hilbert spaces

Web Site for this paper: http://www.math.waikato.ac.nz/~kab

Lecturer for weeks 1-12:

Associate Professor Kevin Broughan, Office G3.22, Tel 838-4423, email kab@waikato.ac.nz

Office Hours during weeks 2-12: normally 3-5pm Thursdays in G3.22.

Lecture times:

  • Tuesday 4.10-5pm in G3.33 (lecture)
  • Thursday 9-9.50am in G3.33 (lecture)
  • Friday 1.10-2pm in G3.33 (Tutorial/Workshop)

    Recommended reading:

    For real analysis: "A first course in mathematical analysis" by Burkill (CUP), "Introduction to real analysis: by Bartle and Sherbert (Wiley), and "Real Analysis and first course" by Gordon (Addison-Wesley). For the metric space sections "Metric spaces" by Copson, (CUP), "Elements of general topology" by Bushaw (wiley) and "Analysis for applied mathematics" by Cheney (Springer).

    Syllabus and On-line lecture notes:

    1. Metric Spaces: page 1, 2, 3, 4, 5.
    2. Subsequences and Cauchy sequences: page 18, 19, 20.
    3. Taylor expansions and their applications: page 38, 39, 40, 41, 42, 43, 44, 45, 46.
    4. Riemann integration I - definition of the integral: page 59, 60, 61, 62.
    5. Uniform continuity: page 7, 8, 9, 10, 11, 12, 13, 14.
    6. Riemann integration II - properties: page 63, 64, 65, 66.
    7. Riemann integration III - continuous and piecewize continuous functions: page 67, 68, 69, 70, 71, 72, 73.
    8. Riemann integration IV - integral of a limit of functions, uniform convergence: page 74, 75, 76, 77, 78, 79.
    9. Exponential and log functions: page 80, 81, 82, 83, 84.
    10. Topology I: page 4, 5, 6, 7, 8, 9.
    11. Metric topologies, closure, interior and boundary: page 10, 11, 12, 13, 14, 15. 16.
    12. Continuity in topological and metric spaces: page 17, 18, 19, 20, 21, 22. 23.
    13. Compactness and the Heine-Borell theorem: page 24, 25, 26, 27, 28, 29.
    14. Completeness and the completion of a metric space. page 30, 31, 32, 33, 34, 35, 36.
    15. Connectness: page 37, 38, 39, 40, 41, 42, 43.
    16. Separable and second countable spaces: page 1, 2, 3, 4, 5.
    17. Ursohn's metrization theorem: 6, 7.
    18. Contraction mappings and applications: 8, 9, 10.
    19. Baire category theorem application to continuous nowhere differentiable functions: 78, 82, 83, 84.
    20. Derivative for functions f: R^n->R^m: page 46, 47, 48, 49, 50.
    21. Chain rule - derivative of composites: page 51, 52, 53, 54.
    22. Jacobian: page 55, 56, 57, 58.
    23. Inverse function theorem: page 59, 60, 61, 62. 63, 64.
    24. Normed spaces: page 65, 66, 67, 68.
    25. Stone-Wierstrass theorem: page 69, 70, 71.
    26. Hilbert spaces: pages 91-94,
    27. The projection theorem: pages 95-98,
    28. Orthonormal sets: pages 99-104,
    29. The isoperimetric theorem: pages 105-107.

    Exercise sheets 2009:

  • Sheet 1: Metric spaces, sequences, Taylor expansions.
  • Sheet 2: Taylor series, Riemann integrable functions.
  • Sheet 3: Integration, uniform convergence.
  • Sheet 4: Uniform convergence.
  • Sheet 5: Metric topology, closure interior and boundary.
  • Sheet 6: Compactness.
  • Sheet 7: Uniform continuity, metric + compact implies separable.
  • Sheet 8: Contraction mappings.
  • Sheet 9: Derivative as a linear map, the chain rule.
  • Sheet 10: Inverse Function Theorem, uniform approximation.

    Assessment:

    There will be 5 assigments each worth 10% and one test worth 50% as follows: You are expected to attend at least 80% of the workshop/tutorials.

    Kevin Broughan

    15th October 2009