MATH3611 is a Mathematics Level III course. See the course overview below.

**Units of credit: **6

**Prerequisites:** 12 units of credit of Level II Mathematics courses with an average mark of at least 70 or higher, including MATH2111 or MATH2011 (Credit), or permission from Head of Department.

**Exclusions:** MATH3570

**Cycle of offering:** Term 2

**Graduate attributes:** The course will enhance your research, inquiry and analytical thinking abilities.

**More information: **This recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains up-to-date timetabling information.

If you are currently enrolled in MATH3611, you can log into UNSW Moodle for this course.

#### Course Aims

Real analysis is a central pillar of modern mathematics, and we will cover its foundations. We start with the concepts of limits and continuity, which are at the core of calculus, and we extend these concepts to quite general situations. The simplest case (metric spaces??) is when there is some way of measuring the distance between two points in the space. In the most important examples, a metric space occurs as a set of functions, so we will look at ways in which one might say that a sequence of functions converges. Taking these ideas one step further, we look at convergence which does not come from a generalized distance function. These are the ideas of point set topology. The course will also include topics such as countability, continuity, uniform convergence, compactness and connectedness.This is not a computational course, but rather one in which you will develop your ability to think and write abstractly, precisely and creatively.

#### Course Description

Limits and continuity are the central concepts of calculus in one and several variables. These concepts can be extended to quite general situations. The simplest of these is when there is some way of measuring the distance between two objects. Some of the most important examples of these `metric spaces' occur as sets of functions, so this course looks at ways in which one might say that a sequence of functions converges. Taking these ideas one step further, we look at convergence which does not come from a generalised distance function. These are the ideas of point set topology. The course includes topics such as countability, continuity, uniform convergence and compactness, as well as an introduction to the core areas of function analysis. This includes the notions of Banach and Hilbert spaces, including Reproducing Kernel Hilbert Spaces which are important in Applied Mathematics, Statistics and elsewhere.