Syllabus
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Course Code: MMATH21-302 Course Name: Functional Analysis |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Normed linear spaces, Banach spaces, finite dimensional normed spaces and subspaces, equivalent norms, compactness and finite dimension, F.Riesz’s lemma. Bounded and continuous linear operators, differentiation operator, integral operator, bounded linear extension, bounded linear functionals, normed spaces of operators, dual spaces with examples. (Scope as in relevant parts of Chapter 2 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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| 2 | Hahn-Banach theorem for normed linear spaces, application to bounded linear functionals on C[a,b], Riesz-representation theorem for bounded linear functionals on C[a,b], adjoint operator, norm of the adjoint operator. Reflexive spaces, uniform boundedness theorem and some of its applications to the space of polynomials and Fourier series. (Scope as in relevant parts of sections 4.1 to 4.7 of Chapter 4 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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| 3 | Strong and weak convergence, open mapping theorem, bounded inverse theorem, closed linear operators, closed graph theorem. (Scope as in relevant parts of sections 4.8, 4.12 and 4.13 of Chapter 4 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) Inner product spaces, Hilbert spaces and their examples, Schwarz inequality, continuity of inner product, orthogonal complements and direct sums, minimizing vector, orthogonality, projection theorem, characterization of sets in Hilbert spaces whose space is dense. (Scope as in relevant parts of sections 3.1 to 3.3 of Chapter 3 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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| 4 | Orthonormal sets and sequences, Bessel’s inequality, series related to orthonormal sequences and sets, total (complete) orthonormal sets and sequences, Parseval’s identity, separable Hilbert spaces. (Scope as in relevant parts of sections 3.4 to 3.6 of Chapter 3 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) Riesz representation theorem for bounded linear functionals on a Hilbert space, sesquilinear form, Riesz representation theorem for bounded sesquilinear forms on Hilbert spaces. Hilbert- adjoint operator, its existence and uniqueness, properties of Hilbert-adjoint operators, self-adjoint, unitary and normal operators. (Scope is as in relevant parts of sections 3.8 to 3.10 of Chapter 3 of ‘Introductory Functional Analysis with Applications’ by E.Kreyszig) |
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