Functional Analysis I 1000-135AF1

Holder's inequalities and Minkowski's inequalities. Completeness and Banach spaces: sequence spaces, the L^p spaces. The notion of a linear functional and its norm, examples. Hilbert spaces, orthonormal bases and orthonormal sets of vectors, examples. The orthogonal projection and the characterization of linear continuous functionals on Hilbert spaces. The notion of a linear operator and its norm. Important example of linear operators: conditional mean, Fourier transform. Hahn-Banach Theorem, the dual space, the space dual to L^p. Adjoint operators on Banach and Hilbert spaces. Banach - Steinhaus Theorem, applications.