Difference between revisions of "Functional Analysis"

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* [[Math 645]]:  Functional Analysis (1?)
 
* [[Math 645]]:  Functional Analysis (1?)
 
* [[Math 646]]:  Functional Analysis (2?)
 
* [[Math 646]]:  Functional Analysis (2?)
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[[Category:Areas]]

Revision as of 12:17, 19 May 2008

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

Linear Analysis

  1. Distribution theory
    • Test functions and distributions
    • Schwartz class and tempered distributions
    • Operations on distributions
  2. Hilbert spaces
    • Riesz representation theorem
    • Projection theorem
    • Lax-Milgram theorem
    • Existence of an orthonormal basis
    • Fredholm alternative
    • Hahn-Banach theorem
  3. Banach spaces
    • Banach-Steinhaus theorem
    • Alaoglu’s theorem
    • Open mapping theorem
    • Bounded inverse theorem
    • Closed graph theorem
    • Baire category theorem
  4. 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

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. Imbedding theorems
  2. Compact imbeddings
    • Poincaré inequalities
    • Rellich-Kondrachov theorem
  3. Morrey's theorem
  4. Caccioppoli inequality
  5. John-Nirenberg inequality
  6. Calderón-Zygmund inequality
  7. Trace theorem
  8. Marcinkiewicz interpolation theorem

Semigroups

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

Courses