Learn the fundamental structures of Functional Analysis.
Get familiar with the main examples of functional spaces, in particular with the theory of Hilbert spaces and Lebesgue spaces.
Get familiar with the basic notions of operator theory. Be able to frame a functional equation in an abstract functional setting.
Lebesgue Measure and Integration.
L^p Spaces.
Basic of Topological Vector Spaces, Normed and Banach Spaces, Linear Operators and linear functionals.
Hilbert Spaces.
Weak topology, Weak * topology, weak compactness.
Applications of Baire Category in Functional Analysis: Uniform Boundedness, Open Mapping, Closed Graph, Inverse Mapping.
Banach and Hilbert adjointness, self-adjointness.
Compact Operators.
Riesz Fredholm spectral theory.
Terence Tao, An introduction to measure theory.. American Mathematical Society, Providence, RI, ISBN: 978-0-8218-6919-2 . 2011.
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations.. Universitext. Springer, New York,. 2011. xiv+599 pp. ISBN: 978-0-387-70913-0
Alberto Bressan, Lecture notes on functional analysis. With applications to linear partial different. Graduate Studies in Mathematics, 143. American Mathematical Society, Providence, RI,. 2013. xii+250 pp. ISBN: 978-0-8218-8771-4.
Michael Reed, Barry Simon, Methods of modern mathematical physics. I. Functional analysis. Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York,. 1980. xv+400 pp. ISBN: 0-12-585050-6.
Stein, Elias M.; Shakarchi, Rami , Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ,. 2005. xx+402 pp. ISBN: 0-691-11386-6