6580- Introduction to Hilbert Spaces (Fall 2013)

MATH 6580
Instructor: Jean Bellissard
Professor of Mathematics and Physics
School of Math, Skiles 132,
Phone: (404) 385-2179 (Math),
Fax: (404) 894-4409
e-mail: jeanbel@math.gatech.edu
Lecture Skiles 269 Tuesday- Thursday 4:35-5:55pm
Office Hours

Tuesday 10:00-10:55am
or by appointment
Skiles 132

Calendar of the Week

Final Exam Thursday December 12th
2:50-5:40pm Skiles 269

(to full Fall calendar )

November 26, 2013
Lecture: Integration with Borel and Spectral Measures No section
December 3-5, 2013
Lecture: The Riesz Theorem, the Spectral theorem No section
Homework #4:
p.399, Exercises: 7;   
p.407, Exercises: 1;   
p.414, Exercise: 4
p.430, Exercise: 2;   
p.436, Exercise: 13, 14;   
p.449, Exercise: 2
Due date:  Thursday November 26th, 2013
Course Outline

Background: Vector spaces, dot products, norms, Cauchy-Schwartz inequality
Contrast the geometry of and other spaces
Complete orthonormal sequences, Fourier series, Bessel's and Parseval's inequality
Projections: closest point projections, linear projections, non-expansive projections, orthogonal projections, and self-adjoint projections
Bounded linear functions, Riesz representation theorem, and the Lax-Milgram theorem
Characterizations of finite dimensional and of self-adjoint, normal, compact, or closed linear operators
A structure for unbounded linear operators, Sturm-Liouville operators
Contraction Mapping Theorem and applications
Some illustrative topics according to students' interest
Normed and Sobolev spaces


Official textbook
   Arch W. Naylor, George R. Sell,"Linear Operator Theory in Engineering and Science", Second edition. Applied Mathematical Sciences, 40. Springer-Verlag, New York-Berlin, 1982
"Linear Operator Theory in Engineering and Science",
Second edition. Applied Mathematical Sciences, 40. Springer-Verlag, New York-Berlin, 1982
In particular Sections 5.12-5.24, 7.1-7.5

Other possible helpful books
      N.I. Akhiezer & I.M. Glazman, Theory of Linear Operators in Hilbert Spaces,
         Dover, New-York (1993),
      Frigyes Riesz & Béla Sz.-Nagy, Functional Analysis, Dover, New-York (1990),

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One compulsory homework will be offered once every 2 or 3 weeks
Each homework will be graded. The homework grade will count for 35% of the final grade

Final Exam

Thursday December 12th,   2:50-5:40pm    Skiles 269    ( to calendar)
An absence to the final be graded 0 (zero). However if the absence is justified (disease, injury, ...), the student must (i) warn the instructor as soon as possible, in any case before the end of the exam week, (ii) bring the documents justifying the absence to the Instructor.
Then the student will receive an I (incomplete) and will be responsible to take action during the following semester to complete his/her curriculum.


90% for an A
Homeworks 35% Grade distribution 80% for a   B
Final 65% 70% for a   C
60% for a   D

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