SYLLABUS

Chapter 1: Hilbert space: geometry, projection on convex sets, orthogonal decomposition, bases, examples. Riesz representation theorem for bounded linear functionals.

Chapter 2: Normed spaces: Young’s inequality, Holder inequality, Minkowski inequality. Spaces of sequences l^p. Spaces of continuous functions C(X). L^p spaces. Equivalent norms. Continuous linear maps and their norms. Finite dimensional normed spaces. Banach spaces. Completion. Product, direct sum and quotient spaces. General criteria for compactness and examples.

Chapter 3: The Baire category theorem. Uniform boundedness. Open mapping theorem. Closed graph theorem. Semicontinuity.

Chapter 4: The Hahn-Banach theorem. Hyperplanes, convex sets and seminorms. Separation of convex sets by hyperplanes. Dual spaces. Duality in sequence spaces, L^p spaces and spaces of continuous functions.

Chapter 5: Weak topologies and duality. Reflexive spaces.

Chapter 6: Theorems involving continuous functions: extension (Tietze, Urysohn), approximation (Stone-Weierstrass), compactness (Ascoli-Arzelà). Some applications.

Chapter 7: Compact operators. Adjoint operators in Hilbert space; self-adjoint operators. Elements of spectral theory for compact, self-adjoint and compact self-adjoint operators. Riesz-Schauder theory. Consequences for Sturm-Liouville problems.

Chapter 8. Regularization. Compactness in L^p spaces.

Chapter 9. An introduction to the theory of unbounded operators and some applications. Closed range theorem.

BIBLIOGRAPHY

Many high quality monographs cover the material of this course and go well beyond. Here is a short selection:

- the classics:

K. Yosida, Functional Analysis, 6th edition, Springer 1980

Kolmogorov e Fomin, Elements of the Theory of Functions and Functional Analysis, Dover 1961

Kantorovitch e Akylov, Functional Analysis, Pergamon 1982

H. Brézis, Analyse Fonctionnelle, théorie et applications, Masson 1983

Martin Schechter, Principles of Functional Analysis, second edition, AMS 2002.

- some more recent and very interesting editions:

W. Cheney, Analysis for Applied Mathematics, Springer 2001

M. Willem, Principes d’Analyse Fonctionnelle, Cassini 2007

In addition there exists excellent material available online. I shall mention the course of Buhler and Salamon (ETH Zurich, 2016), the one by Alan Sokal (UCL 2013), the one by Michael M. Wolf (Munich 2015), and the one of MIT.