Math 241, Fall 2011

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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 1.3-4.

1.2: # 2,3,4,5

1.3: #2,6,8,9,14

1.2

1.3

2 Tue 9/6/10

Section 1.4 – Convergent, bounded and Cauchy sequences. Relations among them.
· Proofs of 1.4-2, 1.4-5, 1.4-6, 1.4-7, 1.4-8

1.4: #1,2,3,4,5 1.4
  Thur 9/8/10 Section 1.4    
3 Tue 9/13/10

Section 1.5 –
· The examples of complete and incomplete metric spaces are important.
· You should be able to give examples of nonconvergent Cauchy sequences.
· Proofs of the theorems in this section required are not required.
Section 1.6 –
· Statement of Theorem 1.6-2
· Examples – Completion of C[a,b] with the d_p metric.

1.5: #5,6,7,8 1.5
  Thur 9/15/10

Section 2.1
· Definition and examples of vector spaces.
· Linear dependence and independence.
· Basis of a vector space. Dimension.
· Subspace. Span of a subset.

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.
· Lemma2.2-9
Section 2.3
· Subspaces of Banach spaces. Theorem 2.3-1
· Infinite series in Banach spaces
· Schauder basis

2.2 #3,4,6,8,9,11,13

2.3 #2,3,5,6,10,11

2.2

2.3

  Thur 9/22/10 Lecture Notes    
5 Tue 9/27/10 Review    
  Thur 9/29/10

Midterm Exam

Solutions

   
6 Tue 10/4/10

Section 2.4
· 2.4-1 Lemma, 2.4-2 and 2.4-3, 2.4-5 Theorems Section 2.5
· Riesz’s Lemma, 2.5-5 Theorem
· 2.5-2 Lemma, 2.5-3 and 2.5-6 Theorems – no proofs

2.4 #1, 2, 8

2.5 #1, 2, 3, 4

2.4

2.5

  Thur 10/6/10

Section 2.6
· Linear operators, domain, range, null space, homeomorphism, examples
· 2.6-9, 2.6-10 Theorems, 2.6-11 Lemma

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
· 2.7-8, 2.7-9 Theorems, 2.7-10 Corollary
· 2.7-11 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
· Normed spaces of linear operators, dual space
· 2.10-1, 2.10-2, 2.10-3 Theorems
· Examples of dual spaces

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
· 3.2-2 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
· 3.3-1 Theorem, 3.3-2, 3.3-5, 3.3-6, 3.3-7 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