Difference between revisions of "Functional Analysis"

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== Chris Grant's Proposed Core Topics List for a 500-level Linear Course==
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== Chris Grant's Proposed Core Topics List for Some Functional Analysis Courses ==
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=== Linear Analysis ===
 
<div style="-moz-column-count:2; column-count:2;">
 
<div style="-moz-column-count:2; column-count:2;">
 
#  Distribution theory
 
#  Distribution theory
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#* Adjoints of bounded/unbounded operators
 
#* Adjoints of bounded/unbounded operators
 
#* Hilbert-Schmidt theorem
 
#* Hilbert-Schmidt theorem
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</div>
 
</div>
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=== Nonlinear Analysis ===
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<div style="-moz-column-count:2; column-count:2;">
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#  Variational problems
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#* Euler-Lagrange equations
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#* Abstract Dirichlet principle
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#* Mazur’s lemma
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#* Tonelli’s theorem
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#* Mountain pass theorem
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#  Fixed-point problems
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#* Contraction mapping principle
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#* Inverse function theorem
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#* Implicit function theorem
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#* Brouwer fixed-point theorem
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#* Schauder fixed-point theorem
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#* Schaefer’s fixed-point theorem
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#* Leray-Schauder fixed-point theorem
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#* Browder-Minty fixed-point theorem
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</div>
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=== Function Spaces ===
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<div style="-moz-column-count:2; column-count:2;">
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# Imbedding theorems
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# Compact imbeddings
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#* Poincaré inequalities
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#* Rellich-Kondrachov theorem
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# Morrey's theorem
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# Caccioppoli inequality
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# John-Nirenberg inequality
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# Calderón-Zygmund inequality
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# Trace theorem
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# Marcinkiewicz interpolation theorem
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</div>
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=== Semigroups ===
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# Hille-Yosida theorem
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# Lumer-Phillips theorem
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# Friedrich's lemma
  
 
== Courses ==
 
== Courses ==
 
* [[Math 645]]:  Functional Analysis (1?)
 
* [[Math 645]]:  Functional Analysis (1?)
 
* [[Math 646]]:  Functional Analysis (2?)
 
* [[Math 646]]:  Functional Analysis (2?)

Revision as of 14:31, 8 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

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