Unit 1

Review of metric spaces, completion of metric spaces, Normed space, Banach space, properties of Normed spaces, Finite dimensional normed spaces and subspaces, Equivalent norms, compactness and finite dimension.

Unit 2

Norm of a linear operator, Bounded and continuous linear operators, Linear functionals, Normed spaces of operators, Dual spaces, Computing Dual of some Banach Spaces.

Unit 3

Inner product space, Hilbert space, Orthogonal complements and direct sums, Orthonormal sets, Bessel inequality , Gram-Scmidt Orthonormalisation, Orthonormal basis, Functionals on Hilbert spaces, Riesz’s theorem, Projection and Riesz representation theorem, Adjoint operator, Self adjoint, Unitary and Normal Operators.

Unit 4

Hahn-Banach theorem, Baire’s Category theorem and Uniform boundedness principle, Open Mapping Theorem, Closed Graph Theorem, Bounded Inverse Theorem, Adjoint Operator, Strong and Weak Convergence, Convergence of sequence of Operators and Functionals.

Unit 5

Spectrum of Bounded Operators, Compact Linear Operators, Spectrum of a Compact Operator, Spectrum and Numerical Range, Spectrum of Bounded Self-Adjoint Operators.

Course outcomes
CO1: Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts.

CO2: Students will have a firm knowledge of real and complex normed vector spaces with their geometric and topological properties. They will be familiar with the notions of completeness, separability, will know the properties of a Banach space and will be able to prove results relating to the Hahn Banach theorems. They will have developed an understanding of the theory of bounded linear operators on a Banach space

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CO3: The Hahn Banach theorem is a central tool in functional analysis. It allows the norm preserving extension of bounded linear functional defined on a subspace of some vector space to the whole space and it also shows that there are enough continuous linear functionals defined on every normed vector space to make the study of the dual space interesting.

CO4: The Uniform boundedness principle is one of the fundamental results in functional analysis. Together with the Hahn-Banach theorem and the open mapping theorem it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators and thus bounded operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The completeness of a norm is exploited to obtain four major theorems, namely the Uniform boundedness principle, the closed graph theorem, the open mapping theorem and the bounded inverse theorem.

CO5: Inner products allow us to think about geometric concepts in vector spaces. Gram Schmidt processes explain how the basis of a normed linear space can be converted into an orthonormal basis. Complete inner product spaces (that is Hilbert spaces) are studied in detail.

CO6: Apply problem solving using functional analysis techniques applied to diverse situations in Physics, Engineering and other mathematical contexts.

Textbook:

1. Kreyszig, Introductory Functional Analysis with Applications, Wiley.
2. B.V. Limaye, Functional Analysis, Wiley.

References:

• M. Thamban Nair, Functional Analysis- A First Course, PHI. A.L.Brown, A.Page, Elements of Functional Analysis, Van Nostrand.