Functional analysis in applied mathematics and engineering

Additional Info

  • ECTS credits: 8
  • Semester: 1
  • University: University of L'Aquila
  • Objectives:

     

    The objective of this course is to provide the main concepts and fundamental methods of Functional Analysis to enable a student to treat various concrete problems arising in applied mathematics and engineering like PDEs, integral equations, best approximation problems, control theory, calculus of variations, numerical analysis.

  • Topics:

     

    Metric & Normed spaces. Uniform convergence and spaces of continuous functions. Compactness in infinite dimension. Lebesgue integral, Lp-spaces. Linear Operators, cornerstone theorems of FA. Dual Spaces, weak convergence. Hilbert Spaces, Fourier series. Adjoint operators. Compact operators. Introduction to spectral theory. Introduction to operator semigroups. Applications to PDE, integral equations, calculus of variations, and numerical analysis.

  • Prerequisites:

     

    Linear Algebra. Complex numbers. Differential and integral calculus of functions of real variables.

  • Books:

     

    • 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
    • Lecture notes provided by the teacher.
    • 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
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