Christopher E. Heil

• Course Syllabus
  
  
• Office Hour: Thursdays at 3:00.
  
  
• Background and supplementary materials:
  • General Reading List
  • Materials used in Math 6337 (Real Analysis I) in Fall 2007
  • Materials used in Math 6338 (Real Analysis II) in Summer 2008
  • Short Reviews:
    • Lebesgue measure
    • Banach and Hilbert spaces
  • Long Reviews:
    • A Basis Theory Primer (Chapter 1 is a review of theorems from functional analysis)
  
  
  
• LECTURE NOTES:
  
  
  • Lecture Notes A: Metrics, norms, inner products, and topology
    • Hints for selected exercises.
  • (Additional reading) Details on equivalence of norms on finite-dimensional spaces
    
    
  • Lecture Notes B: Background reading on Lebesgue measure
    
    
  • Lecture Notes C
    • Operators and Dual Spaces
      • Hints for selected exercises.
      • Zorn's Lemma
    • The dual of L^p
    • Problems on c and c_0
    • Complex Analysis and the Riesz-Thorin Theorem
    • (Background reading) More on operators and functional analysis in Banach spaces
    
    
  • Lecture Notes D
    • Borel and Radon measures on the real line (small updates 9/20/09)
    • Notes on Egoroff's and Lusin's Theorems for Radon measures
      • Older notes on Egoroff's and Lusin's Theorems for Lebesgue measure
    • Proof of the Riesz Representation Theorem I
    • Proof of the Riesz Representation Theorem II
    
    
  • Lecture Notes E
    • Topological vector spaces
      • Proof that L^p[0,1] contains no nontrivial convex open subsets
    • Weakly convergent sequences
    • Banach Limits
    • Abstract TVS
    • Operators on TVS
    • Locally compact TVS
    • Metrizable TVS
    • The Minkowski Functional
    • Seminorms and Normability
    
    
  • Distributions
    • The Inductive Limit Topology on C_c^infinity(R)
    • Completeness of C_c^infinity(R)
    • Background on convolution
    • Background on approximate identities
  
  
  
• HOMEWORK:
  • Homework 1 is due September 22, 2009.
  • Homework 2 is due October 8, 2009.
  • Homework 3 is due October 29, 2009.
  • Homework 4 is due December 3, 2009.