Syllabus:

1. Motivating example: Fourier series.
2. Basics of metric spaces: boundedness, convergence, continuity, compactness.
3. Banach spaces (basic definitions only).  Hilbert spaces. Subspaces. Linear spans.
4. Orthogonal expansions. Bessel’s inequality. The Riesz–Fischer theorem. Orthogonal complements.
5. Fourier series. Fejér’s theorem. Parseval’s formula.
6. Dual space of a normed space. Self-duality of Hilbert space.
7. Linear operators. B(H) as a Banach space. Adjoints. Hermitian, unitary and normal operators.
8. The spectrum of an operator on a Hilbert space. The spectral radius formula.
9. Compact operators. Hilbert–Schmidt operators.
10. The spectral theorem for compact normal operators.
11. Applications to integral equations.
12. Normed spaces, bounded linear operators on a Banach space, dual spaces, Hahn-Banach theorem, Zorn’s lemma. Use of sequence spaces. The Banach space C(X) for a compact space X.
13. Basic measure theory, up to the construction of the Lebesgue measure on the real line. Complex measures. Dominated convergence theorem. Product measures. Fubini theorem. Measurable functions.
14. Definition of spaces of Lebesgue integrable functions, and proof that with the standard norm they form a Banach space. Dual spaces. The Radon-Nikodym Theorem. The conjugate index theorem.
15. The Banach space M(X) of regular Borel measures on a compact space X. Proof that the dual of C(X) is M(X).
16. Introduction to abelian Fourier analysis.
17. Advances of metric spaces: fixed point of contractions, Baire categories, semi-norms.

References

[1]

Nicholas Young. An Introduction to Hilbert Space. Cambridge University Press, Cambridge, 1988. MR # 90e:46001.

[2]

Walter Rudin. Real and Complex Analysis. McGraw-Hill Book Co., New York, third edition, 1987. MR # 88k:00002.

[3]

Erwin Kreyszig. Introductory functional analysis with applications. John Wiley & Sons Inc., New York, 1989. MR # 90m:46003.

[4]

Alexander A. Kirillov and Alexei D. Gvishiani. Theorems and Problems in Functional Analysis. Problem Books in Mathematics. Springer-Verlag, New York, 1982.

[5]

Michael Reed and Barry Simon. Functional Analysis, volume 1 of Methods of Modern Mathematical Physics. Academic Press, Orlando, second edition, 1980.

[6]

A. N. Kolmogorov and S. V. Fomīn. Introductory real analysis. Dover Publications Inc., New York, 1975. Translated from the second Russian edition and edited by Richard A. Silverman, Corrected reprinting.

[7]

A. N. Kolmogorov and S. V. Fomin. Measure, Lebesgue integrals, and Hilbert space. Translated by Natascha Artin Brunswick and Alan Jeffrey. Academic Press, New York, 1961.

Web page: http://v-v-kisil.scienceontheweb.net/courses/math3263.html
Lecture notes HTML: http://v-v-kisil.scienceontheweb.net/courses/math3263m-split.html
Lecture notes PDF: http://v-v-kisil.scienceontheweb.net/courses/math3263m.pdf

---

 

Figure 1: A family of Fejér kernels with the parameter m running from 0 to 9 is on the left picture. For a comparison unregularised Fourier kernels are on the right picture. Animated version is here

---

This document was translated from L^AT_EX by H^EV^EA.