MTH641 Handouts

MTH 641

MODULE NO. 1 TO

(MID TERM SYLLABUS)

THESE ARE JUST SHORT HINT FOR THE PREPARATION OF MTH

Don’t look for someone who can solve your problems, Instead go and stand in front of the mirror, Look straight into your eyes, And you will see the best person who can solve your problems! Always trust yourself.

A gift from Unknown to Juniors VU Mathematics Students INTRODUCTION: MODULE NO. 1

Its applications are in differential equations and numerical analysis, approximation theory and calculus of variations etc.

COURSE OUTCOMES:

To be able to understand basics concepts, principles and methods of functional analysis and its applications.

COURSE OUTLINE: MODULE NO. 2

Topics:

Introduction, Metric space, subspace, Triangle inequality, Axioms of a metric, Sequence space, Space B(A) of bounded functions, Some Inequalities, Ball and sphere, Continuous mapping, accumulation point, Dense set, separable space, Convergence of a sequence, limit, Cauchy sequence, comple teness, Real line, complex plane, Uniform convergence, Discrete metric, Isome tric mapping, isome tric spaces, Homeomorphism, Normed Space, Banach Space, Further Properties of Normed Spaces, Finite Dimensional Normed Spaces and Subspaces, Compactness and Finite Dimension, Linear Operators, Bounded and Continuous Linear Operators, Linear Functional, Linear Operators and Functional on Finite Dimensional Spaces, Normed Spaces of Operators, Dual Space, Inner Product Space, Hilbert Space, Further Properties of Inner Product Spaces, Orthogonal Complements and Direct Sums, Orthonormal Sets and Sequences, Series Re lated to Orthonormal Sequences and Se ts, Total Orthonormal Se ts and Sequences, Legendre Hermite and Laguerre Polynomials, Representation of Functional on Hilbert Spaces, Hilbert Adjoint Operator, Self-Adjoint, Unitary and Normal Operators. MODULE NO. 3

RECOMMENDED BOOKS:

Book Title: Introductory Functional Analysis with Applications

Citation:

Author:

Erwin Kreyszig, John Wiley & Sons. Inc.

Edition:

Publisher: Printed In USA

Book Title: Functional Analysis, Sobolev Spaces and Partial Differential Equations

Citation:

Author:

HaimBrezis. Universitext, Springer.

Edition:

Publisher: sPRINGERsCIENCE+ Business Media,LLC,233 , NY ,USA

Book Title: Introduction to Functional Analysis

Citation:

Author:

Angus E. Taylor, John Wiley & Sons. Inc

Edition:

Publisher: Alpha Science International Limited

Book Title: Elements of Functional Analysis

Citation:

Author:

Robert Zimmer, University of Chicago Lecture Series.

Edition:

Publisher: University of chicago Press