Catalog description: Normed linear spaces and algebras, the theory of linear operators,
spectral analysis. Extended description for Functional Analysis: Math 51305131: Topics in functional analysis, at the choice of the instructor and
students, including the following.
Theory of Banach spaces: duality, reflexivity, weak and weak*
topologies. HahnBanach, BanachSteinhaus, BanachAlaoglu theorems.
KreinMilman theorem. Linear bounded operators on Hilbert spaces:
compact, integral, trace class, Fredholm, HilbertSchmidt, Toepliz,
Volterra. Compact selfadjoint operators with applications to the
classical SturmLiouville theory. Spectral theory of unbounded
selfadjoint and normal operators on Hilbert spaces, quadratic forms.
Examples and counterexamples of selfadjoint operators, and the role
of the Fourier transform. Spectral theory of differential operators
with constant coefficients. Unitary and positivity preserving operator
semigroups, resolvents, Trotter product formula, HilleYosida theorem.
Other topics may include distributions, commutative Banach and C*
algebras, Gelfand transform and its relation to the spectral theorem
for bounded normal operators, Banach algebra L^{1}, Kaplansky density
theorem, GelfandNaimarkSegal construction, introduction to von
Neumann algebras and noncommutative integration. Applications to
probability, quantum mechanics, Kigami's resistance forms and
Dirichlet forms.
Textbook choices:
Functional Analysis by W. Rudin.
A Course in
Functional Analysis by J.B.Conway.
Functional Analysis by P.D. Lax.
Functional Analysis (Methods of Modern Mathematical Physics, Volume 1) by M. Reed and B. Simon.
Spectral Theory of SelfAdjoint Operators in
Hilbert Space by M.S.Birman and M.Z.Solomjak.
Supplementary reading:
Functional Analysis by K. Yosida.
Functional Analysis, Sobolev Spaces
and Partial Differential Equations by H. Brezis.
Lecture notes by A.J.
Wassermann and by V.F.R. Jones.
Lectures on Quantum Mechanics for
Mathematics Students by L.D. Faddeev and O.A. Yakubovskii.
Prerequisites: MATH 5111
Credits: 3
