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MATH806 Functional Analysis
Campus
SNS
Programs
PG
Session
Fall Semester 2016
Course Title
Functional Analysis
Course Code
MATH806
Credit Hours
30
PreRequisutes
Course Objectives
This lecture course aims to introduce students to the basic concepts of Functional Analysis.
Brief Decommention of the Course:
Functional analysis plays an important role in the applied sciences as well as in mathematics itself. Roughly speaking, functional analysis develops the tools from calculus and linear algebra further to the more general setting where one has vector spaces comprising functions or general abstract infinitedimensional vector spaces. Problems from various application areas can then be conveniently posed in this common general set up, and solved using the techniques of functional analysis. The basic objects studied in functional analysis are vector spaces with a notion of distance between vectors, and continuous maps between such vector spaces. This interplay between the algebraic and analytic setting gives rise to many interesting and useful results, which have a wide range of applicability to diverse mathematical problems.
Learning Outcomes:
On successful completion of this course, students will be able to
determine whether a mapping is a metric on a given set;
prove that a given metric space is complete;
determine whether a mapping is a norm on a vector space;
recognize whether a normed space is compact;
know basic properties of bounded and continuous linear operators on normed spaces;
know about the normed space of operators and dual spaces;
prove some results about Hilbert spaces;
recognize unitary, adjoint and normal operators on Hilber spaces;
represent functionals on Hilbert spaces;
know about HahnBanach theorem, open mapping theorem and closed graph theorem.
Detail Content
Metric Spaces: 1.1 Metric Space 1.2 Further Examples of Metric Spaces 1.3 Open Set, Closed Set, Neighborhood 1.4 Convergence, Cauchy Sequence, Completeness 1.5 Examples. Completeness Proofs 1.6 Completion of Metric Spaces
Normed Spaces: Banach Spaces 2.1 Normed Space. Banach Space 2.2 Further Properties of Normed Spaces 2.3 Finite Dimensional Normed Spaces and Subspaces 2.4 Compactness and Finite Dimension 2.5 Linear Operators 2.6 Bounded and Continuous Linear Operators 2.7 Linear Functional 2.8 Linear Operators and Functional on Finite Dimensional Spaces 2.9 Normed Spaces of Operators. Dual Space
Inner Product Spaces. Hilbert Spaces: 3.1 Inner Product Space. Hilbert Space 3.2 Further Properties of Inner Product Spaces 3.3 Orthogonal Complements and Direct Sums 3.4 Representation of Functional on Hilbert Spaces 3.5 HilbertAdjoint Operator 3.6 SelfAdjoint, Unitary and Normal Operators
Some Important Theorems: 4.1 HahnBanach Theorem 4.2 Uniform Boundedness Theorem 4.3 Open Mapping Theorem 4.4 Closed Linear Operators. Closed Graph Theorem
Text/Ref Books
Text books:
Erwin Kreyszig, Introductory Functional analysis with Applications, John Wiley & Sons 1978.
Reference books:
W. Rudin. Real and Complex Analysis; 3rd edition. McGraw Hill, 1987.
Time Schedule
Fall Semester 2014
Faculty/Resource Person
Dr. Rashid Farooq
D.Sc. Mathematics (Kyoto University, Japan)
Discipline: Combinatorial Optimization