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MTH641 : Functional Analysis

Course Overview

Course Synopsis

Functional analysis is a branch of mathematical analysis the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure e.g. inner product norm topology etc. and the linear functions defined on these spaces and respecting these structures in a suitable sense.rnThis course focuses on Norm Space Banach Space Inner Product Space Hilbert Space and their properties.

Course Learning Outcomes

At the end of the course you should be able to

  • Understand and apply the concepts of metric spaces topological spaces convergence and separability.
  • Utilize Cauchy sequences as a fundamental concept in metric spaces understanding their significance in completeness and convergence.
  • Demonstrate knowledge of complete metric spaces recognizing the importance of completeness in mathematical analysis and problemsolving.
  • Explore normed spaces including the definition and properties of norms concept of Banach spaces recognizing the completeness of normed spaces and its significance in mathematical analysis.
  • Study linear operators gaining insight into their properties and applications within normed or metric spaces.Define linear operators between normed spaces and analyze their properties such as boundedness continuity and invertibility.
  • Understand the algebraic dual space understanding about the role of dual spaces in functional analysis and explore the concept of canonical mapping.
  • Knowledge of Inner product space hilbert space and apply these concepts to solve problems related to orthogonality and completeness. Understanding GramSchmidt Orthogonalization process utilizing it to create orthonormal bases in Hilbert spaces.
  • Get knowledge about Hilbert adjoint unitary and normal operators and study their properties.

Course Calendar

1 Course Motivation
2 Introduction and contents of the course
3 Books Recommended for Functional Analysis
4 Metric space Introduction
5 Formal Definition and Axioms of metric Space
6 Subspace of a Metric Space
7 Examples of Metric Space
8 Examples of Metric Sapce (Euclidean and Unitary space)
9 Sequence Space as an example of Metric Space
10 Function space and Discrete MetrIc
11 Sequence space s
12 lp Space and Hilbert Sequence Space
13 Ball and sphere
14 Open set/ball, closed set/ball

15 Neighbourhood of a point & interior of a point
16 Toplogical space
17 Continnous mapping
18 Theorem(Continnous mapping)
19 Accumulation Point
20 Dense set, separable space
21 Examples of separable space
22 lp space as a separable space

23 Sequences and their Convergence
24 Bounded sequence
25 A Lemma related to Boundedness and limit
26 Cauchy sequence, completeness
27 Convergent sequence(Theorem)
28 Theorem related to Closure & Closed set
29 Theorem(Complete Supspace)
30 Continuous mapping Theorem
Assignment 1

31 Completeness of R
32 Completeness of R^n
33 Completeness of l^infinity
34 Completeness of C[a,b]
35 Completion of Metric Spaces
36 Revision(Vector Space)
37 Examples of Vector Space
38 Some Important concepts of vector space

39 Dimension of vector space & related theorem
40 Normed space and Banach space
41 Examples of Normed spaces
42 Unit sphere
43 Subsapce and convergence in Norm spaces
44 Infinite series covergence, Basis and completion theorem
45 Finite Dimensional Normed Spaces
Quiz 1

46 Completeness Theorem
47 Closedness Theorem
48 Equivalent Norms
49 Compactness and Finite Dimension
50 Compactness theorem
51 F-RieszLemma
52 Theorem(Finite Dimension)
53 Continuous mapping theorem and corollary

54 Linear Operator
55 Examples of Linear Operators
56 Range and null space (Theorem)
Quiz 2
57 Inverse Operators
58 Inverse operator (Theorem)
59 Lemma(Inverse of product)
60 Bounded Linear Operator
61 Lemma (Norm)

62 Examples Of Bounded Linear Operators
63 Examples Of Bounded Linear Operators(Matrix)
Mid term examination
64 Theorem Finite Dimension for linear operator
65 Continuity and Boundedness (Theorem)
66 Corollary(Continuity , null space)
67 Types of operators
68 Bounded Linear Extension (Theorem)
69 Linear Functionals

70 Examples of linear functionals
71 More Examples of linear functionals
72 Algebraic Dual space and Canonical mapping
73 Algebraically Reflexive
74 Linear functionals and operators on finite dimension vector spaces
75 Operators on Finite dimensional spaces

76 Zero vector lemma
77 Theorem (reflexivity)
78 Linear Transformation (exercises)
79 Dual Basis
80 Examples of dual spaces

81 More examples of dual spaces
Quiz 3
82 Bounded Linear operators(Theorem)
83 Finite Hilbert Spaces

84 Hilbert space Theorem
85 Norm on Inner Product Space
86 Parallelogram Law
87 Polarization and Appolonious Identity
88 Inner Product Spaces (examples)
89 Inner Product Spaces as Metric Spaces
90 Continuity of inner product Theorem
91 Examples of Inner Product Spaces

92 Orthogonal System(Pythagorean Theorem)
93 Linear Independence Theorem
94 Examples of Orthogonal systems
95 Orthonormal Systems
96 Hilbert Space(Orthogonaliszation Theorem)
Quiz 4
97 Theorem (Riesz and Fischer)
98 The Gram-Schmidt Orthogonalization process
99 Example(Distance and subspace)
100 Annihilators

101 Annihilators (Properties)
102 Direct Decomposition
103 Annihilators(Theorem)
104 Hilbert Adjoint Operator
105 Hilbert Adjoint Operator(Zero Lemma)
106 (Theorem) Properties of hilbert adjoint operator
107 Self Adjoint, Unitary and Normal Operator
108 Self Adjoint, Unitary and Normal Operator(Examples)
109 Self Adjoint, Unitary and Normal Operator(some properties)

110 Self Adjoint, Unitary and Normal Operator(Self adjointness theorem)
111 Theorem(Self adjointness of product)
112 Theorem(Sequences of self adjoint operator)
113 Theorem (Unitary operator)
Final Term Examination

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