----------------------------------------------------------------------

MATH 7132 -  FUNCTIONAL ANALYSIS

Instructor: Jan Mandel
Text: E. Kreyszig: Introductory Functional Analysis with
Applications,  John Wiley & Sons, 1989.

Functional analysis is a combination of linear algebra and analysis.
Linear algebra runs into difficulties for spaces with infinite dimension.
Adding the concept of convergence based on norm or inner products
overcomes these difficulties and resuts in a very useful and simple
theory. Infinitely dimensional spaces occur often as spaces of functions
(such as the space all continuous functions on an interval), hence the
name. 

Functional analysis provides an elegant framework for partial 
differential equations, in particular finite elements (How to compute 
deformation of a structure under a load? Fluid flow? How to measure the error
of an approximate solution... ?) 
Other applications include Fourier series and Fredholm intergral equations
(why do objects vibrate on discrete sets of frequences?), and optimization
(how to minimize an objective function on an infinitely dimensional space?)

Without functional analysis you cannot understand applied mathematics 
from the last half of century!

Prerequisites: Linear algebra, introductory real analysis (5070 or equivalent)
or topology of metric spaces. Lebesgue integration (real analysis)
is useful but not required as a prerequisite.

Topics covered:
Fundamental properties of Banach and Hilbert spaces. Duality and
weak convergence. Applications to differential and integral equations 
and quadrature formulas. Spectral theory of bounded operators, Banach 
algebras. Compact operators and Fredholm alternative. Optimization in
Hilbert spaces.