| CODE | MAT3211 | ||||||||
| TITLE | Functional Analysis: Hilbert Spaces | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 5 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | • Hilbert spaces; • Orthonormal bases; • Least squares approximation; • Spectral theory; • Self adjoint and normal operators. Study-unit Aims: Hilbert spaces are vector spaces equipped with an inner product that induces a complete metric. The theory of Hilbert spaces is concerned with the study of infinite dimensional vector spaces in a way that generalizes familiar concepts in Euclidean spaces. This theory has several applications in other areas of pure and applied mathematics and provides a framework for quantum physics. This study-unit covers the fundamentals of Hilbert space theory and it provides students with a grounding in the theory and techniques used to analyse Hilbert Spaces. Learning Outcomes: 1. Knowledge & Understanding By the end of the study-unit the student will be able to: - Recognize how basic concepts of geometry and linear algebra can be generalised to infinite dimensional Hilbert spaces; - Become familiar with the basic properties of the classical function and sequence spaces regarded as Hilbert spaces; - Appreciate the theory of Hilbert Spaces as a fundamental mathematical tool and recognise its potential applications in other areas of mathematics and physics. 2. Skills By the end of the study-unit the student will be able to: - Calculate the Fourier Series expansion of standard functions; - Relate the Projections Theorem to applications in optimisation problems in other disciplines; - Analyse properties of bounded linear maps between Hilbert Spaces and calculate their spectrum in simple cases; - Prove basic results about Hilbert spaces and apply general results and theorems to solve simple problems in other areas of mathematics and physics. Main text: • 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 | Leads to: MAT5313 & MAT5616 | ||||||||
| STUDY-UNIT TYPE | Lecture | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Beatriz Zamora-Aviles |
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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 2024/5. It may be subject to change in subsequent years. |
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