Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
Course: Functional Analysis Instructor: Chaitanya G K Room: G25 Level: Postgraduate Time: Currently offered |
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Syllabus Past Exams Syllabus: i) Quick review of sequences and series of functions, equicontinuity, Arzela- Ascoli theorem. ii) Normed linear spaces and Banach spaces. Bounded linear operators. Dual of a normed linear space. Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, closed graph theorem. Computing the dual of some well known Banach spaces. Weak and weak-star topologies, Banach-Alaoglu Theorem. The double dual. Lp-spaces and their duality, Weierstrass and Stone- Weierstrass Theorems. iii) Hilbert spaces, adjoint operators, self-adjoint and normal operators, spectrum, spectral radius, analysis of the spectrum of a compact operator on a Banach space, spectral theorem for compact self-adjoint operators on Hilbert spaces. Basics of complex measures and statement of the Riesz representation theorem for the space C(X) for a compact Hausdorff space X. iv) If time permits, some of the following topics may be covered: Sketch of proof of the Riesz Representation Theorem for C(X), Goldsteins Theorem, reflexivity; spectral theorem for bounded normal operators. Suggested Texts : (a) Real and complex analysis, W. Rudin, McGraw-Hill (1987). (b) Functional analysis, W. Rudin, McGraw-Hill (1991). (c) A course in functional analysis, J. B. Conway, GTM (96), Springer-Verlag (1990). (d) Functional analysis, K. Yosida, Grundlehren der MathematischenWissenschaften (123), Springer-Verlag (1980). Evaluation:
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Midterm
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