MATHEMATICS 5102: Linear Mathematics in Infinite Dimensions
Info for the 9:10 a.m. section of SP 2013
Instructor: Dr. Rodica D. Costin
Office: 436 Math Tower
Office hours: MWF after class (10:15-11 a.m.) or
Here are lecture notes for the first two chapters. I am planning
post revised notes as the semester advances.
Topics studied in each lecture and general announcement are posted
Monday Jan. 7 We defined the space
l^2 (Hilbert Spaces notes, Sec. 1.4)
Wed Jan. 9 Hilbert Spaces notes,
Sec. 1.5, 2.1, 2.3, started 2.4.
Homework 1 - due Fri Jan 16. (Actually,
Fri Jan 18th...)
You may work in teams: each team submits only one write-up, signed
by all the members of the team.
Fri Jan. 11 - up to Sec. 3.2
Monday Jan. 14 Orthonormal bases. Revised notes
Further references for Hilbert Spaces:
M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol.
(see the handout).
Wed Jan. 16 Revised
Homework 2 (due Fri Jan 25)
Fri Jan. 18 re-Revised notes
Wed Jan. 23
Fri Jan. 25 Revised
notes for Jan. 23 and 25
Monday Jan. 24 We
finished the chapter on Hilbert spaces, see here
and we started Sturm-Liouville chapter, Sec. 1.1
Wed Jan. 30 Revised
Fri Feb 1st Revised
For Fourier series and integrals, see also the Stanford lectures of
Prof. Osgood, available on Youtube.
See also a handout on Carmen.
Mon Feb 4 Revised
Wed Feb 6 Dirichlet Kernel
Fri Feb 8
The Fourier Theorem;
Smoothness versus decay of Fourier coefficients
Midterm: Wed Feb 20th
Recall the Rayleigh's quotient for
self-adjoint matrices. Warning: disregard the comments
about existence of min/max in finite dimensions; in infinite
dimensions extra care is needed. In the (infinite-dimensional) case
we will study on Monday, R will have a min but not a max.
Mon Feb 11 Completeness. Shannon's
sampling Theorem (compact interval case)
Wed Feb 13 Sturm-Liouville
Oscillation Theorems; the wave equation; the vibrating rod (Sec.
6, 9.1. 9.3 in the Sturm-Liouville notes posted on top of this
Homework 6 (due
Fri Feb 15 Prufer coordinates: existence of eigenvalues
for Sturm-Liouville problems (Sec. 7. in the
Feb 18 Fourier Integral and Examples of Fourier
Wed Feb 20 Midterm
Fri Feb 22 Experiments:
See the Fourier transform!. Wave
- Guest speaker, Prof. Ulrich Gerlach
Mon Feb 25 Wave packets - Prof. Ulrich Gerlach
Homework 7 (due
March 8 - only problems 3,4,8)
Wed Feb 27 Distributions (see also
the hand-out on Carmen)
Fri March 1 Fourier transform of
Mon March 4 Fundamental sequences
for Dirac's delta-function.
the hand-outs on Carmen from Hilbert
& Courant's book. It was written before the
introduction of distributions, and you can see
the physical intuition, beautifully written and
developed. See pages 351-362, and 371-376. Must read!
Wed March 6 Green's
Fri March 8 Uniqueness,
Fredholm Alternative We only discussed
Theorem 8. Please read Theorems 9 and 10 as well, and use them to
solve Homework 8.
A simple example of Green's function, corrected.
8 due March 22
elements of complex functions from last semester.
March 18 - April 12 Helmholtz equation -
lectures by Prof. Gerlach
Translation as exp(d/dx)
Homework 9 due Fri March 29
10 due Fri Apr 5
The revised chapter on
Distributions, Green's function, Adjoint Problems.
Revised notes: Hilbert
Pick up the
final exam on Monday
Apr 22 after class, to be returned in
my office, according the the official schedule seen
scheduled for Tuesday Apr 30 8:00am-9:45am, see Registrar
Final Examination Schedule