UNIVERSITY OF WAIKATO

Department of Mathematics math501-17A

Metric Spaces


The test will be 120 minutes on Friday 16th June 10am to noon, off G3.19.

I will be available to assist students each day Monday - Thursday 12-15th June in G3.22 from 11am- noon.

The list of topics for the test for 2017 is given directly below. It will be essentially a bookwork test, and you will be expected to answer 6 out of 9 given questions. You should prepare to answer "most" questions based on the results below, not just the easier ones.

  1. All definitions and theorem statements covered in lectures
  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. closure interior and boundary in a topological space
  7. metrizable topology is Hausdorff
  8. f continuous then the inverse image of every closed set is closed
  9. closed subsets of a compact space are compact
  10. in a Hausdorff space compact subsets are closed
  11. continuous image of a compact set is compact
  12. continuous image of a connected set is connected
  13. compact metrizatble spaces are sequentially compack
  14. in R^n compact subsets are closed and bounded
  15. if A is connected so is its closure
  16. (a,b) is connected
  17. second countable implies separable
  18. in a metrizable space separable implies second countable
  19. metrizable spaces are normal
  20. compact Hausdorff spaces are normal
  21. contraction mapping theorem

  22. Baire category theorem and application that complete metric spaces are unconuntable
  23. derivative on R^n is unique when it exists
  24. differentiable implies continuous for maps R^n -> R^m
  25. matrix representation for derivatives f:R^n -> R
  26. derivative of a linear map
  27. general chain rule
  28. Inverse function theorem individual part

  29. all norms are equivalent on R^n
  30. norms and metrics on C[0,1] - complete for the sup norm
  31. algebras of functions closed under max and min
  32. Equality of two definitions for the norm of a linear map

  33. Cauchy-Schwartz inequality
  34. parallelogram law and Pythagoras in a Hilbert space
  35. projection theorem in a Hilbert space
  36. projection onto a closed subspace and associated orthononal decomposition into subspaces
  37. A Hilbert space is isomporphic to its dual space
  38. At most a countable number of Fourier coefficients for a given orthonormal set are nonzero
  39. If an orthonormal set is complete then each element of the space has a unique Fourier series representation.

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

Lecturer for weeks 1-12:

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

Office Hours during weeks 2-12: normally 3-4pm Thursdays in G3.22.

Lecture times:

  • Monday 9am in G3.33 (lecture)
  • Tuesday 3.10-4pm in G3.33 (lecture)
  • Thursday 11-11.50am in S.B.03 (Tutorial/Workshop)

    Syllabus and On-line lecture notes:

    1. Metric Spaces: page 1, 2, 3, 4, 5.
    2. Subsequences and Cauchy sequences: pages 1-3,
    3. Taylor expansions and their applications: page 38 updated, 39, 40, 41, 42, 43, 44, 45, 46.
    4. Topology: page 4, 5, 6, 7, 8, 9.
    5. Metric topologies, closure, interior and boundary: page 10, 11, 12, 13, 14, 15. 16.
    6. Continuity in topological and metric spaces: page 17, 18, 19, 20, 21, 22. 23.
    7. Compactness and the Heine-Borell theorem: page 24, 25, 26, 27, 28, 29.
    8. Completeness and the completion of a metric space. page 30, 31, 32, 33, 34, 35, 36. Completion notes from Fairchild and Tulcea.
    9. Every metrizable topological space is normal - in class notes.
    10. Connectness: page 37, 38, 39, 40, 41, 42, 43.
    11. Separable and second countable spaces: page 1, 2, 3, 4, 5.
    12. Ursohn's metrization theorem: 6, 7.
    13. Contraction mappings and applications: 8, 9, 10.
    14. Baire category theorem application to continuous nowhere differentiable functions: 78, 82, 83, 84.
    15. Derivative for functions f: R^n->R^m: page 46, 47, 48, 49, 50.
    16. Chain rule - derivative of composites: page 51, 52, 53, 54.
    17. Jacobian: page 55, 56, 57, 58.
    18. Inverse function theorem: page 59, 60, 61, 62. 63, 64.
    19. Normed spaces: page 65, 66, 67, 68.
    20. Stone-Wierstrass theorem: page 69, 70, 71.
    21. Linear maps (3 pages),
    22. Hahn-Banach theorem (3 pages),
    23. Hilbert spaces: pages 91-94,
    24. The projection theorem: pages 95-98,
    25. Orthonormal sets: pages 99-104,
    26. The isoperimetric theorem: pages 105-107.

    Exercise sheets 2017:

  • Sheet 1: Metric spaces.
  • Sheet 2: Taylor's theorem and its applications.
  • Sheet 3: Topology, closure interior and boundary.
  • Sheet 6: Compactness.
  • Sheet 7: Uniform continuity, metric + compact implies separable.
  • A complete proof of the Lemma on page 35.
  • Examples (i) and (ii) on page 41.
  • Show that (pi,2pi) is an open and closed subset of the rational numbers. Show that the only connected subsets of the rationals are points.
  • Connectness: KB notes Thm 21 p39, Example(i) p41, Prove each point in a topological space is contained in a maximal connected component, these component form a partition of the space and are both open and closed.
  • (a) Take one part of the Urshon metrization theorem and prove it clearly in a way you understand it well. (b) Prove all compact Hausdorff spaces are metrizable.
  • Sheet 8: Contraction mappings.
  • Sheet 9: Derivative as a linear map, the chain rule.
  • Sheet 10: Inverse Function Theorem, uniform approximation.
  • Jacobian: Experiment with a mapping, like f:(x,y)-> 4(x+y,x^2-x^2) say. Find the image of a few unit squares and a rough ratio of their areas. Then compute the Jacobian and find where its determinant is zero. Look at the image of a point where the determinant is zero as the size of the domain square tends to zero. Compare with a point where the determinant is non-zero.
  • Inverse function theorem: find the largest open subset where the differentiable inverse of (x,y)->(4x,y^2-2y+1) exists, and find an expression for it.
  • Taylor expansions: find the Taylor expansion with 3 terms (including the remainder) for a suitable function f:R^2-> R$ and then for $f:R^n-> R$. Hint do it for g:R-> R where g(t)= f(a+th), a and h in R^n.

    Kevin Broughan

    29th May 2017