| CODE | SOR5521 | ||||||||||||
| TITLE | Topics in Topological Vector Spaces and Functional Analysis | ||||||||||||
| UM LEVEL | 05 - Postgraduate Modular Diploma or Degree Course | ||||||||||||
| MQF LEVEL | 7 | ||||||||||||
| ECTS CREDITS | 10 | ||||||||||||
| DEPARTMENT | Statistics and Operations Research | ||||||||||||
| DESCRIPTION | Students taking this study-unit will be assigned a number of topics, selected from the ones below. These would cover areas of direct use to the particular student involved and would make study-time demands commensurate with the number of credits being allotted: The topics listed below are of direct relevance and importance to other advanced topics in Probability, Asymptotic Theory, Stochastic Processes and Optimization. - Topological And Vector Space Settings; - Hilbert Spaces - with special reference to L2; - Banach Spaces - with special reference to Lp ,C[ 0, 1] and spaces of measures; - Operators; - Duality; - Spectral Theory; - Fourier Transform; - Semigroups of Differential Operators. The amount of material selected and the depth and detail at which it will be dealt with will be commensurate with the credit value and the requirements needed for the student to engage effectively in related areas involved in her/his thesis. Study-Unit Aims: Stochastic Processes, Limits in Probability, Statistical Inference and Bayesian Statistics are areas which are dealt with at undergraduate level. As the range of application of these wildly, and widely, used areas in numerous disciplines, delves deeper into mathematical sophistication and abstraction, masters students in Probability, Statistics and OR need a better grounding of some of the topics above. This study-unit has been designed to offer the opportunity and provide adequate space within the academic spread of our masters' studies for our students to top up their knowledge with mathematical repertoire of techniques, conceptual frameworks and deep results relevant to their area of specialization. The offerings here are wide and flexible enough to suit individual needs. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Assemble a strong intuition of how relevant theorems in vector spaces, topology and functional analysis underwrite a number of results and concepts in areas covered while students are working on their thesis; - Comprehend how the leap from finite to infinite dimensions has deeply enriched modern mathematics and the contemporary use of mathematical sophistication in various applied areas; - Be familiar with a number of fundamental properties and results from advanced analysis to which probability, optimization and statistical theory are anchored. 2. Skills: By the end of the study-unit the student will be able to: - Assimilate and interpret correctly the theoretical assumptions underpinning the major theories studied; - Have an appreciation for the staunch mathematical foundations of probability ,statistics and optimization; - Develop the ability to choose the right texts in advanced work and access in a short time to important and fundamental results with reference to mathematical or practical needs as they arise. Main Text/s and any supplementary readings: - Aliprantis, C.D., Border, K.C.(2007) Infinite Dimensional Analysis: A Hitchhiker's Guide, 3rd ed., Springer. - Conway, J. B. (1994) A Course in Functional Analysis, 2nd edition, Springer-Verlag. - Dunford, N., and Schwatrz L., (1958,1963) Linear Operators I, II, Interscience. - Hille, E. and Phillips, R.S., (2008) Functional Analysis and Semigroups, AMS. - Kantorovitz, S., (2006) Introduction to Modern Analysis, Oxford University Press. - Kelley, J.L., (1975) General Topology, Springer-Verlag. - Kirillov, A.A.and Gvishiani, A.D. (1982) Theorems and problems in functional analysis, Springer-Verlag. - Köthe, G. (1983) Topological vector spaces I, Springer-Verlag. - Rudin, W.(1991) Functional Analysis, McGraw-Hill Science. - Schaffer, H., (1999) Topological Vector Spaces , Spriger-Verlag. - Yoshida, K., (1991) Functional Analysis, Springer-Verlag. |
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| STUDY-UNIT TYPE | Independent Study | ||||||||||||
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
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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