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Contents

• V Full version
• 0 Introduction
• 1 Uniform convergence
• 2 Series of functions
• 2.1 Convergence of series
• 2.2 Power series
• 3 Uniform continuity and integration
• 3.1 Uniform continuity
• 3.2 Applications to Riemann integrability
• 3.3 Non-examinable fun*
• 4 ℝⁿ 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 ℝⁿ
• 6.1 Differentiation from ℝ^m to ℝⁿ
• 6.2 The operator norm
• 6.3 Mean value inequalities
• 6.4 Inverse function theorem
• 6.5 2nd order derivatives