PMAT 41393: Functional Analysis

Course Code    : PMAT 41393

Title                   : Functional Analysis

Pre-requisites   : PMAT 21263

 Learning Outcomes:

At the end of this course, the student should be able to

1. use axioms for abstract linear spaces (over the real or complex fields) to discuss examples (and non examples) of abstract linear spaces

2. prove and apply Holder and Minkowski Inequalities and explain the general properties of metric and normed spaces and the relationships between them

3. define convergence, limit and being Cauchy sequence by using functional analysis tools

4. illustrate concepts such as completeness and complement of normed spaces and report on fundamental properties of Banach spaces

5. explain the general properties of inner product and normed spaces and the relationships between them, in particular illustrate concepts such as completeness and complement of inner product spaces.

6. analyze the fundamental properties of Hilbert spaces, the properties of linear operators defined in finite and infinite dimensions and the important applications of these properties

7. define and explain the concepts of continuity and limitation for operator, function, functional, Banach and Hilbert spaces and self-adjoint operators

8. apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.

 Course Contents:

Linear spaces: 

• Linear spaces, Subspaces and convex sets, Quotient space, Direct sums and Projections, Holder and Minkowski Inequalities.

Normed Linear Spaces and Banach Spaces: 

• Normed spaces, Open Balls, Equivalent norms, Open and closed sets, Metric spaces and Metrics Induced by norms, Translation invariant and absolute homogeneity of a norm, Quotient norms, Convergence of a sequence in a normed space, Cauchy sequence, Completeness of normed space, Banach Spaces, Bounded, Totally Bounded, and Compact Subsets of a Normed Linear Space, Finite dimensional normed linear spaces.

Inner Product Spaces and Hilbert spaces: 

• Inner product spaces, Cauchy-Bunyakowsky-Schwarz Inequality, Inner product norm, Polarization and Parallelogram Identity, Completeness of Inner product spaces, Hilbert spaces, Orthogonality, Best approximation or nearest point, orthonormal basis.

Bounded Linear Operators and Functionals: 

• Linear Operator, Kernal and Null spaces, Inverse Linear Operator, bounded linear operators, Operator norms, uniformly operator convergent and strongly operator convergent, Bounded linear functionals, Open Mapping Theorem, Banach’s Theorem.

 Method of Teaching and Learning : A combination of lectures, tutorial discussions and presentations.

 Assessment :  Based on tutorials, tests, and end of course examination.

 Recommended Reading:

1. Teschl, G. (2020). Topics in linear and nonlinear functional analysis. American Mathematical Society.

2. Halmos, P.R. (1996). Finite Dimensional Vector Spaces, Springer.

3. Markin, M.V. (2018). Elementary Functional Analysis, de Gruyter.

4. Pinchuck, A. (2011). Functional Analysis Notes, Rhodes University.

5. Madox, I.J. (1992). Elements of Functional Analysis, Cambridge University Press.

6. Jain, P.K., Ahuja, O.P. & Ahmad, K. (2nd Ed., 2010). Functional Analysis, New Age Science Limited.

7. Kreyszig, E. (2007). Introductory Functional Analysis with Applications, John Wiley, New York.