Entry requirements

60 credits in mathematics. Basic Topology is recommended.

Learning outcomes

The course provides the theoretical basis of results of previous calculus courses and of deeper studies in pure mathematics. In order to pass, the student should be able to

• describe the construction and properties of real numbers;
• explain the theoretical basis of differential and integral calculus including the formulation of central theorems and the main features of their proofs;
• explain the basic theory of metric spaces and its application to function spaces;
• apply the theory to solve mathematical problems including the construction of simple proofs.

Content

Definitions and properties of real numbers. Cauchy sequences, upper and lower limits, open and closed sets, compact sets, Heine-Borel lemma. Continuous functions. Differentiable functions: mean-value theorem and its consequences, Taylor series. Riemann integral. Metric spaces and their topology. Sequences and series of functions: uniform convergence, equicontinuous families, Arzelà-Ascoli theorem. Stone-Weierstrass theorem. Banach's fixed point theorem and applications. Inverse and implicit function theorems, rank theorem.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course.

Other directives

The course cannot be included in higher education qualification together with Real Analysis (1MA088), 5 credits.