UNIVERSITY OF WAIKATO

Department of Mathematics math501-18A

Metric Spaces

The test will be 120 minutes on Thursday 7th June 2pm to 4pm, off G3.19.

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 defined

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. assuming Urysohn's lemma prove his metrization theorem for second countable normal spaces
22. contraction mapping theorem

23. Baire category theorem and application that complete metric spaces are unconuntable
24. derivative on R^n is unique when it exists
25. differentiable implies continuous for maps R^n -> R^m
26. matrix representation for derivatives f:R^n -> R
27. derivative of a linear map
28. general chain rule
29. Inverse function theorem individual part either one can assume Df(a)=I or solving a minimization problem to show an inverse exists

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

34. Cauchy-Schwartz inequality
35. parallelogram law and Pythagoras in a Hilbert space
36. projection theorem in a Hilbert space
37. projection onto a closed subspace and associated orthononal decomposition into subspaces
38. A Hilbert space is isomporphic to its dual space
39. At most a countable number of Fourier coefficients for a given orthonormal set are nonzero
40. Every Hilbert space has a complete orthonormal set
41. If an orthonormal set is complete then each element of the space has a unique Fourier series representation
42. The set of compact linear operators on a Banach space is closed in the operator norm
43. An operator on L^2([0,1],C) given by an integral with a real bounded symmetric kernel is self adjoint and all of its eigenvalues are real.

### 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.
27. The spectral theorem for compact operators - notes to be provided.

## Exercise sheets 2018:

• Sheet 1: Metric spaces.
• Sheet 2: Taylor's theorem and its applications.

Kevin Broughan

28th May 2018