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.

- All definitions and theorem statements covered in lectures
- min{d,M} for M>0 is a metric equivalent to d
- a convergent sequence in (X,d) is Cauchy. In R, Cauchy implies convergent
- in R, sufficient conditions via Taylor for a local minimum, point of inflection
- Taylor expansions derived
- closure interior and boundary in a topological space
- metrizable topology is Hausdorff
- f continuous then the inverse image of every closed set is closed
- closed subsets of a compact space are compact
- in a Hausdorff space compact subsets are closed
- continuous image of a compact set is compact
- continuous image of a connected set is connected
- compact metrizatble spaces are sequentially compack
- in R^n compact subsets are closed and bounded
- if A is connected so is its closure
- (a,b) is connected
- second countable implies separable
- in a metrizable space separable implies second countable
- metrizable spaces are normal
- compact Hausdorff spaces are normal
- contraction mapping theorem
- Baire category theorem and application that complete metric spaces are unconuntable
- derivative on R^n is unique when it exists
- differentiable implies continuous for maps R^n -> R^m
- matrix representation for derivatives f:R^n -> R
- derivative of a linear map
- general chain rule
- Inverse function theorem individual part
- all norms are equivalent on R^n
- norms and metrics on C[0,1] - complete for the sup norm
- algebras of functions closed under max and min
- Equality of two definitions for the norm of a linear map
- Cauchy-Schwartz inequality
- parallelogram law and Pythagoras in a Hilbert space
- projection theorem in a Hilbert space
- projection onto a closed subspace and associated orthononal decomposition into subspaces
- A Hilbert space is isomporphic to its dual space
- At most a countable number of Fourier coefficients for a given orthonormal set are nonzero
- 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**

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.

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

Kevin Broughan

29th May 2017