Your lecture notes will cover all the material
(except for those results assigned for self-study) in the syllabus. The occasional chapter references
are to more extensive explanations, and refer to Rudin's Principles.
Aspects of the theory of sets, relations and functions
The natural numbers, the principle of mathematical induction, Peano arithmetic
Number systems, the rational numbers, fields, ordered fields and the "usual order" on the rationals
The least upper bound property, the real line, construction of the real line (Chapter 1: Appendix)
The Archimedean property of the real line, complex numbers, Euclidean spaces, the Cauchy–Schwarz inequality
and associated inequalities (the section The Complex Field in Chapter 1)
Countable and uncountable sets, cardinality
Metric spaces, open and closed sets in metric spaces and associated concepts
Compact sets, the characterisation of compact subsets of Euclidean spaces
Cantor sets, perfect sets (the section Perfect Sets in Chapter 2), connected sets
Sequences and convergence
Subsequences, subsequential limits, the limits of special sequences (the section Some Special Sequences in Chapter 3)
Cauchy sequences, completeness, sufficient conditions for completeness
Infinite series and their convergence, criteria for convergence
Convergence tests for non-negative series,
criterion for summability of the series of pth powers
The extended real number system, limits at infinity, upper and lower limits
Absolute convergence, the Ratio and Root Tests, conditional convergence,
using convergence of series to derive limits of sequences
Topics listed up to this point comprised the syllabus of the mid-term examination.
The limit of a function: various equivalent definitions, the algebra of limits
Continuous functions
Continuity and compactness, attainment of extreme values, uniform continuity
Continuity and connectedness, the intermediate-value theorem, and applications
Left- and right-hand limits (as in the section Discontinuities in Chapter 1)
Differentition in one variable
Lagrange's mean value theorem and its applications
The chain rule for differentiation in one variable
Taylor's theorems: the approximation theorem and the mean value theorem
Integration: motivation, the Riemann integral, and a characterisation of Riemann integrability
Riemann integrability of continuous functions, functions with discontinuities
The first and second Fundamental Theorems of Calculus
Techinques: the change-of-variable formulas for the Riemann integral, integration by parts
Sequences of functions, examples and motivations for uniform convergence
Uniform convergence and Riemann integration
The normed linear spaces Cb(X;R) and Cb(X;C) and their completeness
Equicontinuity, compact sets in C(K;R): the Arzela–Ascoli Theorem
The Weierstrass approximation theorem (Note: we omit the Stone–Weierstrass theorems.)
Vector-valued functions in several real variables: limits and continuity
Matrix representations of linear transformations, the matrix norm
The meaning of differentiation of functions in several variables: the total derivative
Partial derivatives, matrix representation of the total derivative
The chain rule for the total derivative, applications of the chain rule