FUNCTIONAL ANALYSIS NOTES Fall 2007 - Spring 2008
©Gabriel Nagy

DISCLAIMER; These are my handwritten notes, scribbled either before or after I covered the topic in class. Inevitably there will be tons of typos.God bless you if you can read my handwriting :) At some point I hope to type all this stuff.

1. Topology Review on Convergence: Ultrafilters & Nets  (Incomplete! This is a continuation of the discussion started in first class, for which I don't have notes!)
2. Brief Metric Topology Review (includes Baire's Thm. and pre-compactness)
3. Hahn-Banach Theorem (with several applications, inclding Banach Limit Thm.).
   
4. Typed!  "Strange" Limits. (This note brings some clarification on #3 above. At some point Cesaro's Theorem is quoted. For more information on that result read here. Updated 9/25/07: corrected typos. THANK YOU, BOB BURCKEL!)
5. (NEW 9/19/07) Typed! Topological Vector Spaces: The new set is broken into three separate files
   • TVS I: Basic Theory (updated 10/16/07.: added a couple of Exercises; corrected a few typos)
     
   • TVS II: Constructing New Spaces Out of Old Ones:(updated 10/1/07: removed one Exercise)
   • TVS III: Finite Dimensional Spaces (updated 9/25/07: corrected typos. THANK YOU, BOB BURCKEL!)
   • TVS IV: Completeness and Metrizability (modified 11/16/07: added Bourbaki Embedding Lemma; renumbered some Exercises; updated 2/11/08: added Theorem 1; updated 2/14/08: streamlined Thm.1)
6. Typed! Convexity "Warm-ups": 
   • "Warm-up" I: Convex Sets and Convex Hulls (updated 10/17/07; added several cool Exercises!)
     
   • "Warm-up" II: The Minkowski Functional (added 9/20/07)
   • "Warm-up" III: The Importance of Being .... Open and Convex (added 9/20/07; modified 9/22/07: added one exercise)
     
7. Locally Convex Vector Spaces:
   
• LCVS I: Basic Local Theory (added 9/26/07; modified 10/16/07: added one Exercise)
  
• LCVS II: "Natural" Constructions in the Locally Convex Category (updated 10/1/07)
• LCVS III: The Metric Point of View (modified 10/31/07: divided old LCVS III into two parts; reorganized & added some stuf; modified 11/02/07: added one remark after Thm-Def 1; updated 11/04/07: corrected some typos)
• LCVS IV: Examples (created 10/31/07: this is half of old LCVS III)
• LCVS V: Linear Continuous Maps and Topological Duals (expanded 10/14/07; now includes Krein-Milman, Milman, and Alaoglu-Bourbaki Theorems; modified 10/15/07: streamlined sub-section C; modified 10/31/07 with new numbering - this is the old LCVS IV)
8. Duality Theory (created 10/31/07, by splitting the old LCVS V)
   
   • DT I: Basic Theory (created 10/31/07: half of old LCVS V; modified slightly 11/01/07)
     
   • DT II: Advanced Theory (posted 2/15/08)
     
9. Banach Spaces (created 11/25/07)
   
   • BS I: Normed Vector Spaces (created 11/25/07; updated 1/23/08)
   • BS II: Elementary Banach Space Theory (created 1/24/08; modified 2/5/08 modified 2/25/08: added some stuff on vector-valued \ell^p spaces))
   • BS III: Banach Spaces of Continuous Functions (uploaded 1/24/08; updated 1/29/08: pretty much complete, but I need to re-number Theorems and Exercises; updated 2/7/08; updated 2/9/08: added more stuff; updated 3/4/08: some minor notation changes)
   • BS IV: Banach spaces of Measurable Functions (uploaded 3/4/09: contains a lot of stuff worth reading!)
     
   • BS V: A Closer Look at Weak and Weak* Topologies (uploaded 2/10/09)
   • BS VI: Vector-valued Calculus (uploaded 4/12/08; updated 4/26/08: added "easy" Fubini)
     
10. Hilbert Spaces
    
    • HS I: Basic Properties (posted 4/25/08)
    • HS II: Applications to Measure and Integration Theory (uploaded 3/31/08: includes Radon-Nikodim stuff)
    • HS III: Linear continuous  maps (coming soon)
    • HS IV: Finite Dimensional Spaces (coming soon)
      
11. Banach Algebras and Basic Operator Theory
    
    • BA I: Elementary Properties (posted 5/8/08; includes a detailed discussion on L^1(G), which was covered in bits and pieces seceral times in class; updated 5/9/08: added a small sub-section at the end)
    • BA II: Invertibility and Spectrum
      
    • BA III: Abelian Banach Algebras (coming soon)
    • BA IV: Holomorphic Functional Calculus (coming soon)
    • BA V: C*-algebras and Continuous Functional Calculus (coming soon)
    • BA VI: Spectral Measures and Borel Functional Calculus (coming soon)
12. Unbounded Operators on Hilbert Spaces
    
    • UB I:.  Elementary Properties (coming soon)
    • UB II:  Stone's Theorem (coming soon)
      
13. Compact and Fredholm Operators
    
    • CF I: Elementary Properties (coming soon; here are a few handwritten notes, that cover part of this section; TYPO ON PAGE 2: it should be X / X_{n-1} instead of X / X_n)
      
    • CF II: Fredholm Operators in Hilbert spaces (coming soon)
    • CF III: Schatten-von Neumann Classes (coming soon)
    • CF IV: Applications to Representation Theory for Compact Groups (coming soon)