Course Information

Time: MWF 9:10-10
Location: BLOC 123
Instructor: Dean Baskin
Office: BLOC 614B
Office hours: M 1:30-2:30, F 10-11
Syllabus

Announcements

  • Homework 4 had two errors in its original version: In problem 2, the denominator should be in terms of the metric rather than an absolute value. In problem 4a, the series should converge to the inverse of I-T
  • Office hours on Friday, February 7 and Monday, February 10 will not be held as planned. Instead, I will hold office hours 11-12 on Thursday, February 6 and 1:30-2:30 on Tuesday, February 11.
  • There was an error in the first problem of homework 3; compactness should be countable compactness. Another error was in problem 2(b): the word compact was missing. Both have been changed online.
  • I am open to moving my Monday office hours to a different time on Monday (within constraints of my schedule). If they currently do not work for you, please let me know times that do work for you on Mondays.
  • I will hold additional office hour time on Tuesday, January 28 from 4pm to 4:45pm.

Lecture topics

  • January 13: Syllabus highlights. Definition of topology, countability axioms.
  • January 15: Separability axioms. Continuity. Product topology.
  • January 17: Nets. Statements of Urysohn Lemma and Tietze extension theorem. Beginning of proof. No office hours on Monday, January 20.
  • January 22: Proof of Urysohn Lemma. Compactness.
  • January 24: Locally compact Hausdorff spaces. Statement of Tychonoff theorem.
  • January 27: Proof of Tychonoff theorem. Arzela–Ascoli and proof.
  • January 29: Locally compact version of Arzela–Ascoli, statement and beginning of proof of Stone–Weierstrass.
  • January 31: Proof of Stone–Weierstrass.
  • February 3: End of Stone–Weierstrass, definition of normed vector space and Banach space.
  • February 5: Examples of Banach spaces, bounded linear between normed spaces.
  • February 7: Bounded linear functionals, Hahn–Banach (real version).
  • February 10: Hahn–Banach (complex version), consequences.
  • February 12: More consequences of Hahn–Banach, Baire category statement.
  • February 14: Baire category theorem, Open mapping theorem.
  • February 17: Closed graph theorem, principle of uniform boundedness.
  • February 19: Exam.
  • February 21: Interesting subclasses of operators, including compact operators.
  • February 24: Topological vector spaces.
  • February 26: Locally convex topological vector spaces, Frechet spaces.
  • February 28: Weak topologies, topologies on dual spaces.
  • March 3: Operator topologies, Hilbert spaces.
  • March 5: Hilbert spaces.
  • March 7: Hilbert spaces.
  • March 17: Hilbert spaces, separation by hyperplanes.
  • March 19: Separation by hyperplanes.
  • March 21: Separation by hyperplanes.
  • March 24: Kakutani’s theorem, Lp spaces.
  • March 28: Lp spaces.
  • March 31: Lp spaces.
  • April 2: Lp spaces
  • April 4: The dual of Lp.
  • April 7: Chebyshev inequality, Schur’s test.
  • April 9: Riesz–Thorin interpolation.
  • April 11: Riesz–Thorin interpolation
  • April 14: Riesz representation.
  • April 16: Riesz representation.

Homework