MTH641 Official Handouts

Module No. 1

Introduction:

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.

Module No. 2

Course Outline:

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, completeness, Real line, complex plane, Uniform convergence, Discrete metric, Isometric mapping, isometric 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 Related to Orthonormal Sequences and Sets, Total Orthonormal Sets 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

Module No. 4

In functional Analysis we shall study more general “spaces” and “Functions” defined on them.

In functional analysis we shall study more general “spaces” and “functions” defined on them.

The given below is the real line

,x y  

The distance function with two points x,yor usual metricon real line is d x y x y

Say we have two points -1 and 3 and if we want to measure distance between 3 and -1 then

( 1,3) d  is same as (3, 1) d

3 ( 1)

For example: If we want to measure the distance between 1.8 and -3.5 then

(1.8, 3.5) 1.8 ( 3.5)

1.8 3.5 5.3 x x d x x x -1

1.8 -3.5