This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered will normally include the following, with some additions andvariations each year

• Topological Spaces - continuity, homeomorphisms, convergence, Hausdorff spaces, compactness, connectedness, path connectedness.
• Measure and Integration - Lebesgue outer measure, measurable sets and integration, Lebesgue integral and basic properties, convergence theorems, connection with Riemann integration, Fubini's theorem, approximation theorems for measurable sets, Lusin's theorem, Egorov's theorem, Lp spaces.
• Hilbert Spaces - elementary properties such as Cauchy Schwartz inequality and polarization, nearest point, orthogonal complements, linear operators, Riesz duality, adjoint operator, basic properties or unitary, self adjoint and normal operators, review and discussion of these operators in the complex and real setting, applications to L2 spaces and integral operators, projection operators, orthonormal sets, Bessel's inequality, Fourier expansion, Parseval's equality, applications to Fourier series.
• Calculus in Euclidean Space - Inverse and implicit function theorems.

This is an Honours Pathway Course. It emphasises mathematical rigour and proof and develops modern analysis from an abstract viewpoint.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of advanced analysis such as topology and Lebeque integration and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of advanced analysis techniques
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from advanved analysis
4. Apply problem-solving using advanced analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.

Assessment will be based on:

• 4 or 5 assignments (total 50%; LO 1-4)
• Mid semester and final exams (50%) Details will be discussed in class and provided online.

36 lectures, tutorials by arrangement

A mark of 60 or more in MATH2320.

MATH3021

Completion of MATH2405 is strongly recommended.