Suggestion: Work ahead. If you have looked at the section and the corresponding problems before the lecture, then you will usually have two opportunities to ask questions (and at least a couple days) before we move on.

Tentatively: Homework will be due on the Monday after it is assigned.

   Date    Chapter.Section/Topics   or   Page/Problems

M   May 17  Introduction: Abstract Spaces.  Normed Vector Spaces.
            HW: 1.7.1,3(a),4(a,c),5,7,9,11,13,14,15
W    19     We talked about vector spaces and were in the middle 
            of proving that l^2 ("little-ell-two") is a vector space.  
            We got H"older's inequality, and today, we'll see what we 
            can do with that.  Also, we should get to Norms.
            HW:  1.7.16,17,18,19, 22
F    21     Bases, norms, convergence, open balls, open and closed sets.
M    24     Basic basic functions and topology, sequential compactness
            HW: 1.7.27,28,30,32,34
W    26     Compactness and Completeness
            HW: 1.7.36,41,42,43,44
F    28     Uniform continuity and various norms on C^0[0,1]
            Series; relation between absolute summability and completeness.
M    31     Holiday
W  June 2   The space of bounded linear mappings
            HW: 1.7.45,46,47
F     4     More on B(X,Y); extensions of linear maps
M     7     Uniform Boundedness Principle; contraction mapping
            HW: 1.7.51,52,53,54
W     9     (continued topics from last lecture) Integration (intro)
F    11     Measure on R (i.e., the length of general sets)
M    14     Integration, L^p, examples, and convergence theorems
            HW: 2.16.10,13,14,15,16,19,20,25,36
W    16     L^p spaces, other examples
F    18     Midterm Exam
M    21     Hilbert Spaces; inner products, examples, 
            weak derivatives and Sobolev space
W    23     Weak convergence
F    25     Orthonormal bases, Bessel's inequality and Parseval's identity
M    28     Generalized Fourier Series
            HW: 3.8.2,4,6,7,8,9,10,11,12,13
W    30     Completeness of Fourier Basis (Fejer's Theorem)
            HW: 3.8.16,18
F  July 2   Fejer's Theorem (continued)
M     5     Holiday
W     7     Orthogonality in Hilbert Space
            HW: 3.8.19-22,24,28,29,34,35
F     9     Orthogonality continued; Problems; convexity
M    12     Problems; Riesz Representation
            HW: 3.8.36-38,46,47,49,50,52-54,56
W    14     Riesz Representation continued; Operators
F    16     The Spectral Theorem (basic version)

List of possible presentation/special topics:

Bilinear Forms (section 4.3)  Van Nguyen and Sajid Saleem
Spectral Geometry (section 4.9) Hassan Jaleel
Fredholm Alternative and PDE (chapter 6; section 6.5)  Jianfang Liu and 
                                                     Abdul Basit Memon
Wavelets (chapter 8)  Steve Conover and Abdul Naveed
Optimization (chapter 9) Tansel Yucelen and Martin Meuller and 
                         Sebastian Hilsenbeck and Jean De Montaudouin

Unbounded Operators (section 4.11) Syed Hassan
Heisenberg Uncertainty Principle (sections 7.3-4) Ben Sirb
Weierstrass' Theorem  Benjamin Reames
More on quantum mechanics  Kartik Iyer



(This is actually a list of many acceptable topics.  
Find an interesting result, and build a 20 minute presentation 
around that.  Or you can choose some other topic that is relevant 
and interesting.)


M    19     Applications to PDE
            HW: 4.12.1-10,56-58
W    21     Bilinear Forms (4.3a) and Heisenberg Uncertainty Principle

F    23     Bilinear Forms (4.3b), 
            Optimization (Yucelen)
M    26     Spectral Geometry (4.9) and 
            Optimization/Euler Lagrange Equation (Mueller) 
W    28     Unbounded Operators and 
            Wavelets 
F    30     Weierstrass' Theorem and PDE
            Optimization/Shannon Sampling Theory (section 9.8, Hilsenbeck 
            and Montaudouin) 
            Quantum Mechanics


M  Aug 2    Final Exam Period 11:30-2:20
            More presentations?  

Some (easy) things you should have learned/be able to do 
after taking this course:

Show that any two norms on a finite dimensional space are equivalent.
Know what the Minkowski Inequalities are.
Know little l^p and big L^p spaces and the linear functionals on each.
Give basic examples: Convergence in L^p but not pointwise
                     Pointwise but not in L^p
Dual space:  Definition of norms; equivalent formulations
             Riesz Representation
If Y is Banach, then B(X,Y) is Banach.  (Should be able to prove.)
Prove the Cauchy-Schwarz in any inner product space.
Explain why L^2 and l^2 (any any other Hilbert space) are isomorphic.
Calculate the integral of the characteristic function of the rationals 
or the irrationals (or a Cantor set).
The standard Fourier bases (sine, cosine, exponential)
What is a quadratic form, what is a bilinear form
What does the spectral theorem say
If you have distinct eigenvalues, show they are orthogonal.