Math 6310 Functional analysis
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 limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function spaces
and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary and so on, operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. This course will provide students with a fundamental introduction to the subject, more precisely
- Metric spaces; Continuous maps; Normed linear spaces; Banach spaces via operators and functionals: Hilbert spaces and their operators.
The assumed knowledge will be real analysis and Lebesgue integration theory at the undergraduate level.
- Functional analysis, second edition by Walter Rudin, McGraw-Hill, Inc. 1991.
- Integral and functional analysis by Jie Xiao, Nova Science Publishers, 2008.
- Elementary functional analysis by Barbara MacCluer, Graduate Texts in Mathematics, Springer, 2009.