Overview
Basic properties and foundational results of infinite dimensional normed vector spaces (function spaces) and linear operators acting on them.
Basic properties and foundational results of infinite dimensional normed vector spaces (function spaces) and linear operators acting on them.
About this paper
| Paper title | Functional Analysis |
|---|---|
| Subject | Mathematics |
| EFTS | 0.0833 |
| Points | 10 points |
| Teaching period | Semester 2 (On campus) |
| Domestic Tuition Fees ( NZD ) | $620.00 |
| International Tuition Fees | Tuition Fees for international students are elsewhere on this website. |
- Restriction
- MATH 401-412
- Limited to
- BA(Hons), BSc(Hons), PGDipArts, PGDipSci, MA (Thesis), MSc, MAppSc, PGDipAppSc, PGCertAppSc
- Contact
Mathematics 400-level programme coordinator: Dr Fabien Montiel fabien.montiel@otago.ac.nz
- Teaching staff
- Paper Structure
Fundamentals/Review
- Topology, metric, norm, convergence, continuity.
- Completeness, Banach spaces, Lp-spaces, inequalities.
Completeness
- Baire Category’s Theorem.
- Separability, Schauder basis.
Compactness: Theorems of
- Heine-Borel, Bolzano-Weierstrass.
- Stone-Weierstrass.
- Arzela-Ascoli.
Linear Operators
- Examples, dual spaces, operator spaces.
- Adjoints, Closed Range Theorem.
Foundational Theorems
- Principle of Uniform Boundedness.
- Open mapping, Inverse mapping, and Closed Graph Theorem.
- Theorem of Hahn-Banach.
Reflexivity and weak convergence
- Properties of strong and weak convergence.
- Theorem of Banach-Alaoglu.
Semigroups of linear operators
- Bounded senigroups, groups.
- Unbounded operators, generators, resolvents.
- Theorem of Hille-Yosida.
- Theorem of Lumer-Phillips.
- Teaching Arrangements
18 lectures (50 minutes each).
Tutorials: Weekly drop-in sessions for help with assignments.
Assessment: 3 significant written assignments (100% of final mark)
- Graduate Attributes Emphasised
Critical Thinking, Interdisciplinary Perspective, Lifelong Learning.
View more information about Otago's graduate attributes.- Learning Outcomes
On completion of the study of this paper, students are expected to:
- Understand fundamental approaches in Functional Analysis.
- Know important properties of function spaces and linear operators.
- Understand how methods from real analysis and linear algebra can be used to rigorously proof theorems in functional analysis.
- Understand the power of working in the appropriate Banach space to solve a wide array of problems, particularly in PDEs and stochastic processes.