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Major topics:\r\n(1) Norms and seminorms, Banach and Fréchet spaces, Hahn-Banach and separation theorems, Uniform Boundedness Principle, Open Mapping and Closed Graph Theorems.\r\n(2) Dual spaces, uniformly convex and reflexive spaces, Radon-Nikodým Theorem and the dual of Lp, Banach-Alaoglu’s Theorem, Mazur’s Theorem, adjoint operators.\r\n(3) Compact operators, compactness of adjoint, spectral theory and Fredholm alternative for compact operators, application to differential equations.\r\n(4) Hilbert space and operators on Hilbert space, Lax-Milgram Theorem, Fourier series, spectral theorem for compact self-adjoint operators, application to differential equations.","Order":1},{"ID":"2e5f053b-8835-4692-be49-41f07234cfff","Title":"Prerequisites","Description":"MA4211 and {MA4262 or MA5205}, or departmental approval","Order":2},{"ID":"6e5f053b-8835-4692-be49-41f07234cfff","Title":"Syllabus","Description":"SYLLABUS\r\n
\r\n\t\t\t\tAll students should download a detailed outline from the workbin. \r\n\t\t\t\t \r\n\t\t\t\t1. A crash course on general topology \r\n\t\t\t\t2. Topological vector spaces \r\n\t\t\t\t3. Hilbert space \r\n\t\t\t\t4. Hahn-Banach and separation theorems \r\n\t\t\t\t5. Baire Category Theorem and consequences \r\n\t\t\t\t6. Dual space and reflexivity \r\n\t\t\t\t7. Weak and weak* topologies \r\n\t\t\t\t8. Compact operators \r\n\t\t\t\t \r\n\t\t\t\tThere is no assigned text for the module. For your information, I list some of the sources that I have consulted while compiling the outline. \r\n\t\t\t\t \r\n\t\t\t\t1. Linear operators, part I, by Dunford and Schwarz. Wiley. \r\n\t\t\t\t2. A course in functional analysis, by Conway. Springer GTM 96. \r\n\t\t\t\t3. Functional analysis, by Rudin. McGraw-Hill. \r\n\t\t\t\t4. Functional analysis, by Lax. Wiley. \r\n\t\t\t\t5. Topology, by Dugundji. Allyn and Bacon. (For prerequisites in topology.) \r\n\t\t\t\t \r\n\t\t\t\tI will also upload an extended version of my outline, including some proofs, from time to time. | \r\n\t\t
\r\n\t\t\t\tThe grade for this class will be determined from 3 components: \r\n\t\t\t\tThere will be two graded homework assignments during the semester which together will account for 20% of the grade. \r\n\t\t\t\tThere will be one midterm exam which will account for 30% of the grade. We will do the midterm exam during one of the class periods and let you know the precise date at least two weeks in advance. \r\n\t\t\t\tThere will be one final exam at the end of the semester accounting for 50% of the grade. | \r\n\t\t