Time and Location:

• The lectures take place on Mondays and Wednesdays from 11:00am to 12:15pm in 24-112.

• The first lecture will be on March 6, 2017.

Contact Information:

• Office: 27-314.

• E-mail: ponge [dot] snu [at] gmail [dot] com.

Evaluation:

• Evaluation will be based on homework. There will be about 5-6 homework assignments throughout the semester.

• Students who have missed too many classes and have not turned in all their homework assignments will get an F-grade.

Course Outline:

• Part 1: Locally Convex Spaces.

  1. Locally convex spaces.

  2. The Hahn-Banach theorems.

  3. Applications of Baire Category Theorem: Uniform Boundedness Principle, Open Mapping Theorem, Closed Graph Theorem.

  4. Weak topologies. Banach-Alaoglu Theorem.

• Part 2: Distribution Theory.

  1. Distributions.

  2. Distributions with compact support.

  3. Tensor products of distributions.

  4. Convolution and regularization of distributions.

  5. Tempered distributions and Fourier transform.

  6. Sobolev spaces and elliptic operators (if time is permitted).

Prerequisites:

• Real analysis (Banach and Hilbert spaces, measure theory, Lebesgue's integral, Lp-spaces).

• Elementary topology (including neighborhoods, and topologies defined by a system of neighborhoods).

References:

• Main References:

  1. Conway, J.: A Course in Functional Analysis. Graduate Texts in Mathematics, Springer, 2nd Edition, 1994 (link).

  2. Friedlander, G.; Joshi, M.: Introduction to the theory of distributions. Cambridge University Press, 2nd Edition, 1999 (link).

• Supplementary References:

  1. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, 2011 (link).

  2. Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Dover Publications, 2006 (link).