Functional Analysis I
Semester II (PS02CMTH24)
old course
Syllabus
| Unit-I | Inner product spaces, normed linear spaces, Banach spaces, examples of inner product spaces, Polarization identity, Schwarz inequality, parallelogram law, uniform convexity of the norm induced by inner product, orthonormal sets, Pythagoras theorem, Gram-Schmidt othonormalization, Bessel’s inequality, Riesz-Fischer theorem. Hilbert spaces, orthonormal basis, characterization of orthonormal basis, separable Hilbert spaces. |
| Unit-II | Uniqueness of best approximation from a convex subset of inner product space to a point, orthogonality and best approximation, Gram matrix and its applications, existence and uniqueness of best approximation from a convex subset of a Hilbert space to a point, continuity of a linear mapping, projection theorem and Riesz representation theorem, reflexivity of a Hilbert space. Unique Hahn-Banach extension theorem, weak convergence and weak boundedness. |
| Unit-III | Bounded operators, equivalence of boundedness and continuity of an operator, boundedness of the operator associated to an infinite matrix, adjoint of a bounded operator, properties of adjoint, relations between zero space and the range of operators, normal, unitary and self-adjoint operators, examples, characterizations and results pertaining to these operators, positive operators and generalized Schwarz inequality. |
| Unit-IV | Spectrum, eigenspectrum, approximate eigenspectrum, definition and characterization, spectrum of a normal operator, numerical range, relations of numerical range and different spectra, spectral theorem for a normal/self-adjoint operator on a finite dimensional Hilbert space, compact operators, properties of compact operators, Hilbert-Schmidt operator and its properties, spectrum of a compact operator, spectral theorem for a compact self-adjoint operator. |
Reference Books
- Limaye B. V., Functional Analysis, New Age International Publ. Ltd., New Delhi, 1996.
- Simmons, G.F., Introduction to Topology and Modern Analysis, McGraw-Hill Co., Tokyo, 1963.
- Thumban Nair, Functional Analysis: A First Course, Prentice-Hall of India, New Delhi, 2002.