MATH6212 Analysis 2: Topology, Lebesgue Integration and Hilbert Spaces

Offered By	Department of Mathematics
Academic Career	Graduate Coursework
Course Subject	Mathematics
Offered in	First Semester, 2011 and First Semester, 2012
Unit Value	6 units
Course Description	This course is intended both for continuing mathematics students 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 include: 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 as Banach spaces Maximal Functions Vitali covers, Lebesgue differentiation, and density results Hilbert Spaces Elementary properties such as Cauchy Schwartz inequality and polarization Orthogonal complements Linear operators Riesz duality Applications to L2 spaces and integral operators Projection operators Orthonormal sets Bessel's inequality Fourier expansion Parseval's equality Applications to Fourier series Note: Graduate students attend joint classes with undergraduates but will be assessed separately.
Learning Outcomes	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 advanced analysis 4. Apply problem-solving using advanced analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.
Indicative Assessment	Assessment will be based on Assignments and Final Exam after class discussion
Workload	36 lectures, tutorials by arrangement
Course Classification(s)	AdvancedAdvanced courses are designed for students having reached 'first degree' level of assumed knowledge, which provide a deep understanding of contemporary issues; or 'second degree' and higher levels of knowledge; or for transition to research training programs. and SpecialistSpecialist courses are designed for students having reached 'first degree' level of assumed knowledge, which provide for the acquisition of specialist skills; or 'second degree' and higher level of knowledge; or for transition to research training programs; or knowledge associated with professional accreditation.
Areas of Interest	Mathematics
Eligibility	Bachelor degree; with second year Mathematics.
Requisite Statement	Second year Mathematics is required.
Consent Required	Please contact MATHSadmin@maths.anu.edu.au for consent to enrol in this course.
Programs	Master of Mathematical Sciences and Master of Mathematical Sciences
Academic Contact	MATHSadmin@maths.anu.edu.au

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