# Part IB - Analysis II

## Lectured by N. Wickramasekera, Michaelmas 2015

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Uniform convergence
- 2 Series of functions
- 3 Uniform continuity and integration
- 4 ℝ
^{n}as a normed space - 4.1 Normed spaces
- 4.2 Cauchy sequences and completeness
- 4.3 Sequential compactness
- 4.4 Mappings between normed spaces
- 5 Metric spaces
- 5.1 Preliminary definitions
- 5.2 Topology of metric spaces
- 5.3 Cauchy sequences and completeness
- 5.4 Compactness
- 5.5 Continuous functions
- 5.6 The contraction mapping theorem
- 6 Differentiation from ℝ
^{m}to ℝ^{n}