UNIVERSITY OF WAIKATO

Department of Mathematics math501-18A

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Metric Spaces*

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

Kevin Broughan

16th February 2018