Difference between revisions of "Functional Analysis"

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== Chris Grant's Proposed Core Topics List for Some Functional Analysis Courses ==
 
== Chris Grant's Proposed Core Topics List for Some Functional Analysis Courses ==
=== Linear Analysis ===
 
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#  Distribution theory
 
#* Test functions and distributions
 
#* Schwartz class and tempered distributions
 
#* Operations on distributions
 
#  Hilbert spaces
 
#* Riesz representation theorem
 
#* Projection theorem
 
#* Lax-Milgram theorem
 
#* Existence of an orthonormal basis
 
#* Fredholm alternative
 
#* Hahn-Banach theorem
 
# Banach spaces
 
#* Banach-Steinhaus theorem
 
#* Alaoglu’s theorem
 
#* Open mapping theorem
 
#* Bounded inverse theorem
 
#* Closed graph theorem
 
#* Baire category theorem
 
# Normed linear spaces
 
#* Spectral radius of bounded operators
 
#* Riesz-Schauder theorem
 
#* Analyticity of resolvent operator
 
#* Nonemptiness of spectrum
 
#* Adjoints of bounded/unbounded operators
 
#* Hilbert-Schmidt theorem
 
 
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=== Nonlinear Analysis ===
 
=== Nonlinear Analysis ===
 
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<div style="-moz-column-count:2; column-count:2;">
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=== Function Spaces ===
 
=== Function Spaces ===
 
<div style="-moz-column-count:2; column-count:2;">
 
<div style="-moz-column-count:2; column-count:2;">
 +
#  Distribution theory
 +
#* Test functions and distributions
 +
#* Schwartz class and tempered distributions
 +
#* Operations on distributions
 
# Imbedding theorems
 
# Imbedding theorems
 
# Compact imbeddings
 
# Compact imbeddings
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== Courses ==
 
== Courses ==
 +
* [[Proposed Linear Analysis Course]]
 
* [[Math 645]]:  Functional Analysis (1?)
 
* [[Math 645]]:  Functional Analysis (1?)
 
* [[Math 646]]:  Functional Analysis (2?)
 
* [[Math 646]]:  Functional Analysis (2?)
  
 
[[Category:Areas]]
 
[[Category:Areas]]

Revision as of 09:19, 3 June 2008

Chris Grant's Proposed Core Topics List for Some Functional Analysis Courses

Nonlinear Analysis

  1. Variational problems
    • Euler-Lagrange equations
    • Abstract Dirichlet principle
    • Mazur’s lemma
    • Tonelli’s theorem
    • Mountain pass theorem
  2. Fixed-point problems
    • Contraction mapping principle
    • Inverse function theorem
    • Implicit function theorem
    • Brouwer fixed-point theorem
    • Schauder fixed-point theorem
    • Schaefer’s fixed-point theorem
    • Leray-Schauder fixed-point theorem
    • Browder-Minty fixed-point theorem

Function Spaces

  1. Distribution theory
    • Test functions and distributions
    • Schwartz class and tempered distributions
    • Operations on distributions
  2. Imbedding theorems
  3. Compact imbeddings
    • Poincaré inequalities
    • Rellich-Kondrachov theorem
  4. Morrey's theorem
  5. Caccioppoli inequality
  6. John-Nirenberg inequality
  7. Calderón-Zygmund inequality
  8. Trace theorem
  9. Marcinkiewicz interpolation theorem

Semigroups

  1. Hille-Yosida theorem
  2. Lumer-Phillips theorem
  3. Friedrich's lemma

Courses