Comprehensive Examination Syllabus for Applied Analysis (Math 611)



(1) Normed linear spaces, subspaces, finite dimensional normed linear spaces. 
(2) Bounded linear transformations, The principle of uniform boundedness,
open mapping and closed graph theorems.
(3) Continuous linear functionals, The Hahn-Banach (continuous extension and
separation) theorems, dual and reflexive spaces, dual of L_p, 
(4) Weak and weak* topologies, Banach-Alaoglu theorem. 
(5) Completely continuous 
(compact) operators, Fredholm theory, elements of spectral theory.
(6) Hilbert spaces, projection theorem, Riesz representation theorem, reflexivity.
(7) Orthonormal sets, Bessel's inequality, Parseval's identity.
(8) Adjoint of an operator, self-adjoint, normal, and projection operators,
invertibility and spectrum, spectra of special operators. 
(9) Applications to optimization and PDEs.

Miscellaneous topics (can be covered depending on time, but not asked in the exam):
Topological vector spaces, distributions, Krein-Milman theorem, Banach algebras.



References:

[1.] Foundations of Modern Analysis by A. Friedman, Chapters 4, 5, and 6.
[2.] Real and Complex Analysis by W. Rudin, Chapters 4 and 5.
[3.] Functional Analysis by W. Rudin, Chapters 3 and 4.


Additional references:

[1.] Introduction to Topology and Modern Analysis by G.F. Simmons.
[2.] Linear Operator Theory in Engineering and Science by A.W. Naylor and G.R. Sell.
[3.] Introduction to Functional Analysis by A.E. Taylor.
[4.] Functional Analysis by P. Lax