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Week  Date  Content  Homework  Solutions 
1  Tue 8/30/11  Section 1.1 Definition and examples of metric spaces.  1.1: #6,7,8,13  1.1 
Thur 9/1/10  Section 1.2 Further examples of metric spaces. Section 1.3 – The topology of metric spaces · Open sets, closed sets, balls, neighborhoods. · Interior point, accumulation point. · Closure. Dense subsets. Separable metric spaces. · Continuous mappings. Proof of Theorem 1.34. 
1.2: # 2,3,4,5 1.3: #2,6,8,9,14 

2  Tue 9/6/10  Section 1.4 – Convergent, bounded and Cauchy sequences. Relations among them. 
1.4: #1,2,3,4,5  1.4 
Thur 9/8/10  Section 1.4  
3  Tue 9/13/10  Section 1.5 – 
1.5: #5,6,7,8  1.5 
Thur 9/15/10  Section 2.1 
2.1 #4,5,6,10  2.1  
4  Tue 9/20/10  HW 1 due  Exercises from Chapter 1 Section 2.2 · Defintion of normed spaces, Banach spaces. · Lemma2.29 Section 2.3 · Subspaces of Banach spaces. Theorem 2.31 · Infinite series in Banach spaces · Schauder basis 
2.2 #3,4,6,8,9,11,13 2.3 #2,3,5,6,10,11 

Thur 9/22/10  Lecture Notes  
5  Tue 9/27/10  Review  
Thur 9/29/10  
6  Tue 10/4/10  Section 2.4 
2.4 #1, 2, 8 2.5 #1, 2, 3, 4 

Thur 10/6/10  Section 2.6 
2.6 #1, 2, 3, 7, 8, 12, 13, 14, 15  2.6  
7  Tue 10/11/10  Section 2.7 · Bounded linear operators, norm, examples · 2.78, 2.79 Theorems, 2.710 Corollary · 2.711 Theorem – no proof 
2.7 # 1, 2, 3, 5, 6, 7, 8, 9  2.7 
Thur 10/13/10  Section 2.8 · Linear functionals, definition, norm, algebraical dual space 
2.8 #2, 12, 15  2.8  
8  Tue 10/18/10  Section 2.10 
2.10 #4, 8, 9, 10, 11  2.10 
Thur 10/20/10  
9  Tue 10/25/10  Section 2.10 Section 3.1 · Inner product spaces, orthogonality, · Parallelogram law, polarization identity 
3.1 #1, 2, 3, 4, 6, 8, 9  3.1 
Thur 10/27/10  
10  Tue 11/1/10  HW 2 due Exercises from Chapter 2  
Thur 11/3/10  Study Guide for Exam 2 revised 11/7/11  
11  Tue 11/8/10  Midterm Exam 2  
Thur 11/10/10  Section 3.2 · Schwarz inequality, triangle inequality · 3.22 Lemma · Isomorphism of inner product spaces · Subspaces 
3.2 #4, 5, 7, 8, 9  
12  Tue 11/15/10  Section 3.3 · Convex sets · Minimizing vector, orthogonal projection · 3.31 Theorem, 3.32, 3.35, 3.36, 3.37 Lemmas · Orthogonal complement, direct sum 
3.3 #1, 2, 3, 5, 6, 7, 8, 9  
Thur 11/17/10  
13  Tue 11/22/10  
Thur 11/24/10  
14  Tue 11/29/10 


Thur 12/1/10 


15  Tue 12/6/10 


Thur 12/8/10  
16  Tue 12/13/10  
Thur 12/15/11 
Final Exam 3:00pm 