MATH 680A, Course topics

  1. Metric spaces
    1. Distances, Isometries (D1.1-3)
    2. Lp-distances on Rn and spaces of functions: inequalities of Hölder and Minkowski
    3. Metric space topology (D1.4-8,11,13)
    4. Cauchy sequences, completeness (D1.14)
    5. Contraction principle. Application to differential equations (Picard-Lindelöf's theorem)
    6. Compact spaces, compact sets (D1.16-17)
    7. Arzela-Ascoli theorem. Application to differential equations (Peano's theorem)
  2. Normed spaces
    1. Normed spaces, operators (F5.1)
    2. Linear functionals. Dual space. Weak convergence (F5.2)
    3. Hahn-Banach theorem (F5.2, Z21.1)
    4. Gauge functions of convex sets
    5. Separation of convex sets (Z21.2)
    6. Complex H-B theorem. Second dual and reflexive spaces (F5.2)
    7. The Baire category theorem. Uniform boundedness principle (F5.3)
    8. The open mapping theorem. The closed graph theorem (F5.3)
  3. Hilbert spaces
    1. Inner product, definitions. Cauchy-Schwarz inequality (F5.5)
    2. Fourier series; Bessel's inequality, Parseval's identity. Riesz-Fischer's theorem. (F5.5)
    3. Lemma on Orthogonal Projection. Riesz representation theorem (F5.5; LN)
    4. Application to a boundary-value problem (LN)
  4. Lp spaces
    1. Measure spaces. Approximation by simple functions. Completeness (F6.1)
    2. Hölder's inequality and its converse (F6.1-2)
    3. The dual of Lp (F6.2)
    4. Radon-Nikodym's theorem (F6.2, problem 18.)
  5. Linear operators and spectral theory
    1. Bounded linear operators, resolvent, spectrum
    2. Transpose of an operator.
    3. Symmetric operators on a Hilbert space (Z14.2)
    4. Hilbert-Schmidt theory (Z14.2)
    5. The Fredholm alternative (Z14.3)
    6. Application to integral equations (Z14.4)
    7. Application to eigenvalue problem for a Sturm-Liouville operator (Z14.5)

D: Dieudonne, Foundations of Modern Analysis, Academic Press, 1969.
F: Folland, Real Analysis: Modern Techniques and their Applications, second edition, Wiley 1999.
Z1: Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, Appl. Math. Sci, vol. 108, Springer, 1995.
Z2: Zeidler, Applied Functional Analysis: Main Principles and Their Applications, Appl. Math. Sci, vol. 109, Springer, 1995.
LN: Lecture notes from MATH 592A, February 2008.