| CODE | MAT3210 | ||||||||
| TITLE | Functional Analysis: Normed Spaces | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 5 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - Linear spaces - Sub-linear functionals, semi norms, The Hahn-Banach Theorem - The space Lp for p 鈭� [1, 鈭瀅 - Normed spaces - Definition and Examples - Finite-dimensional normed spaces - Quotients of normed spaces - Linear operators - Approximating Lp functions (p < 鈭�) by continuous functions and continuous functions by polynomials - The Dual Space - The dual Space of Lp (p < 鈭�) - Applications of the Baire Category Theorem: Open Mapping Theorem; Banach Inversion Theorem; Closed Graph Theorem; Banach-Steinhaus Uniform Boundedness Theorem. Study-Unit Aims: Classically, functional analysis is the study of infinite dimensional vector spaces of functions, and linear operators between them. These spaces can be collectively and abstractly modelled by normed spaces. These are vector spaces that are equipped with a metric that 'respects the linear structure'. The goal of the course is to help students who pursue advanced studies in mathematics and related fields to lay a solid foundation in functional analysis. Learning Outcomes: Knowledge and understanding: At the end of this unit the students will have a firm knowledge of real and complex normed vector spaces, with their geometric and topological properties. They will become familiar with the classical Lp spaces, will know the general properties of Banach spaces and operators between them. Skills: Understand the fundamental properties of normed spaces and operators between them, and be able to apply them. Main Texts and Supplementary Readings: Main Texts: - Course notes (E. Chetcuti). - Muscat J., Functional Analysis, Springer, 2014. - Kreysig E., Introductory Functional Analysis, Wiley, 1989. Supplementary Reading: - Rudin W., Functional Analysis, Tata McGraw-Hill, 1973. - Kolmogorov A.N. and Fomin S.V., Elements of the Theory of Functions and Functional Analysis, Dover, 1957. |
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| ADDITIONAL NOTES | Follows from: MAT3217 Leads to: MAT5313 |
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| STUDY-UNIT TYPE | Lecture | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Emanuel Chetcuti |
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The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
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