Differential Analysis: 18.155, Fall 2001

Richard Melrose

• General information

  Original Syllabus or as Postscript or Acroread.

• Old lecture notes -- these are notes from an old (and somewhat different) version of the course:
  • Lectures.ps [Postscript]
  • Lectures.pdf [Acrobat]

• Problem sets and solutions

  
  Anonymous quiz handed out in class on first day: Postscript, Acroread
  1. First problem set (due September 11).
     
     • HW1.ps [Postscript]
     • HW1.pdf [Acrobat]
     • HW1S.ps [Postscript, with solutions]
     • HW1S.pdf [Acrobat, with solutions]
  2. Second problem set (due September 18).
     
     • HW2.ps [Postscript]
     • HW2.pdf [Acrobat]
     • HW2S.ps [Postscript, with solutions]
     • HW2S.pdf [Acrobat, with solutions]
     • HW2S [with solutions]
  3. Third problem set (due September 25).
     
     • HW3.ps [Postscript]
     • HW3.pdf [Acrobat]
     • HW3S.ps [Postscript, with solutions]
     • HW3S.pdf [Acrobat, with solutions]
  4. Fourth problem set (due October 2).
     
     • HW4.ps [Postscript]
     • HW4.pdf [Acrobat]
  5. Fifth problem set (due October 16).
     
     • HW5.ps [Postscript]
     • HW5.pdf [Acrobat]
  6. Sixth problem set (due October 23; postoned to October 25).
     
     • HW6.ps [Postscript]
     • HW6.pdf [Acrobat]
     • HW6S.ps [Postscript, with solutions]
     • HW6S.pdf [Acrobat, with solutions]
  7. Seventh problem set (due October 30).
     
     • HW7.ps [Postscript]
     • HW7.pdf [Acrobat]
  8. Eighth problem set (due November 6).
     
     • HW8.ps [Postscript]
     • HW8.pdf [Acrobat]
  9. Ninth problem set (due November 20).
     
     • HW9.ps [Postscript]
     • HW9.pdf [Acrobat]
  10. Tenth problem set (due December 4).
      
      • HW10.ps [Postscript]
      • HW10.pdf [Acrobat]
  11. Eleventh, last and optional, problem set (due December 11).
      
      • HW11.ps [Postscript]
      • HW11.pdf [Acrobat]

• Lectures

  [H]=Hormander's book.
  1. September 6: Introduction.
  2. September 11: Differentiability, Test functions, [H] Sect 1.2.
  3. September 13:
     • Finished proof of [H], Theorem 2.1.4 (sequential continuity).
     • Restriction and localization, [H] Theorem 2.2.1, Theorem 2.2.4.
     • Inclusion of continuous functions, [H] Theorem 1.2.4.
     • Muliplication by smooth functions, differentiation.
  4. September 18:
  5. September 20:
     • Distributions with support at a point.
     • Homogeneous functions smooth outside the origin as distributions, order not in {-n, -n-1, ....}
     • Two definitions of homogeneity and their equivalence.
     • Fundamental solution of the Laplacian.
  6. September 25:
     • Positive extended real numbers.
     • Volume of an open set.
     • Outer measure of general sets.
     • Countable subadditivity.
     • Lebesgue measurability.
     • (Caratheodory's theorem) The $\sigma$-algebra of Lebesgue measurable sets.
     • (Caratheodory's theorem) Lebesgue measure and completeness.
     • Borel measures and Lebesgue measurabiliy of open sets.
  7. September 27:
     • Review, countability of Lebesgue measure
     • Borel sets on the extended line.
     • Measurability of extended functions.
     • Integral of simple functions.
     • Integral of non-negative measurable extended function.
     • Zero integral
     • Monotone convergence
     • Existence of simple approximants.
     • Linearity of integral and integrability of measurable functions.
     • The space L¹(E).
  8. October 2:
     • Fatou's lemma.
     • Dominated convergence.
     • Almost-everywhere equality.
     • The spaces $L^p(A)$ where A is measurable.
     • Minkowski and Hölder inequaliies.
     • Completeness of the $L^p$ norms.
  9. October 4 -- short lecture
     • Completeness of $L^p(U)$ again.
     • Tempered test functions.
     • Tempered distributions.
  10. Ocober 9: No lecture (holiday)