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Course Contents

Fundamentals of nor med linear spaces: Nor med linear spaces, Riesz lemma, characterization of finite dimensional spaces, Banach spaces. Bounded linear maps on a nor med linear spaces: Examples, linear map on finite dimensional spaces, finite dimensional spaces are isomorphic, operator norm. Hahn Banach theorems: Geometric and extension forms and their applications. Three main theorems on Banach spaces: Uniform bounded ness principle, divergence of Fourier series, closed graph theorem, projection, open mapping theorem, comparable norms. Dual spaces and ad joint of an operator: Duals of classical spaces, weak and weak* convergence, Banach Alaoglu theorem, ad joint of an operator. Hilbert spaces: Inner product spaces, orthonormal set, Gram Schmidtorthonormalization, Bessels inequality, Orthonormal basis, Separable Hilbertspaces. Projection and Riesz representation theorem: Orthonormal complements, orthogonal projections, projection theorem, Riesz representation theorem. Bounded operators on Hilbert spaces: Adjoint, normal, unitary, self ad joint operators, compact operators, eigen values, eigen vectors, Banach algebras. Spectral theorem: Spectral theorem for compact self adjoint operators, statement of spectral theorem for bounded self ad joint operators.