MATH2701 Principles of Analysis

This topic includes: Basics of mathematical logic, quantifiers; natural numbers, Peano axioms, principle of mathematical induction; rational numbers, real numbers, completeness axiom, abstract characterisation of real numbers, algebraic numbers, complex numbers, other number systems; elements of naive set theory, cardinal numbers, countable sets, continuum; sequences, limit of a sequence and its properties, Bolzano and Weierstrass theorems, Cauchy sequences; series, convergence tests; continuity, local and global properties of continuous functions, uniform continuity; metric spaces, complete metric spaces, open, closed and compact sets; differentiation, differentiation theorems, the mean value theorem and its applications, Taylor formula; the Riemann integral and its properties, fundamental theorem of calculus, integrable functions, the Riemann-Stieltjes integral; sequences and series of functions, power series, radius and interval of convergence, analytic functions, uniform convergence, differentiation and integration of power series; introduction to functional analysis: linear spaces, normed spaces, complete normed spaces, inner product spaces.

This topic aims to introduce basic facts of analysis which form foundation of modern mathematics and which every student with degree of Bachelor of Mathematics must know. One of the aims of this topic is to make students well acquainted with rigorous proofs.

1. Have a rigorous understanding of different number systems used in mathematics and to understand a special place and significance of the real number system
2. Know basics of naive set theory and mathematical logic
3. Be acquainted with basics of topology of metric spaces
4. Understand proofs and be able to prove mathematical statements
5. Have knowledge and understanding of differential and integral calculus of one rigorous on rigorous level
6. Have a knowledge of the very basic functional analysis