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Functional Analysis
Programme
1. Introduction. Fourier series on the circle.
a. The wave and heat equations.
b. Basic properties on Fourier series. Convolutions, Approximations of the unity.
c. Convergence in L2. Riemann-Lebesgue theorem.
2. Banach spaces
a. Metric, normed and inner product spaces, Complete metric spaces. Baire’s theorem.
b. Compact and separable spaces.
c. Bounded lineal operators.
d. Linear functionals. Hahn-Banach theorem.
e. The dual and bidual space. Reflexive spaces.
3. Fundamental theorems
a. Banach-Steinhaus theorem.
b. Open mapping and closed graph theorems.
Extra. Weak and *weak topologies. Banach-Alaoglu theorem.
4. Hilbert spaces
a. Orthogonal complements. Direct sums.
b. Riesz’ representation theorem.
c. Orthogonal systems and their Fourier coefficients.
5. Spectral theory for lineal operators.
a. Adjoint operators.
b. The resolvent and the spectrum of an operator.
c. Compact operators. Diagonalization for these operators on separable Hilbert spaces.
Lista Ejercicios
Referencias
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References
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Abuabara T., Lesmes J., Elementos de Análisis funcional, Universidad de los Andes, Ediciones Uniandes, 2010
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Stein E. Shakarchi R. Fourier Analysis. An Introduction, Princeton university press, 2003
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Stein E. Shakarchi R. Real Analysis. Measure Theory, Integration, and Hilbert Spaces, Princeton university press, 2007
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Meise R., Vogt D., Introduction to Functional Analysis, Oxford graduate text in Mathematics, 1997.
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Taylor A., Lay D. Introduction to Functional Analysis. Second edition, John Wiley & Sons 1979.
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Reed M., Simon B. Methods of Mathematical Physics, Vol.1, Functional Analysis. Academic Press 1980.
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Brezis H. Functional analysis, Sobolev spaces and partial differential equations. Springer Universitext 2011.
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