Wassermann A.J. Functional Analysis

The Archimedeans, Cambridge University, 1999. — 58 p.Metric spaces.
Preliminaries.
Compactness and completeness.
Normed spaces.
Continuity properties of the algebraic operations.
Completeness.
The uniform norm.
The space l2.
The lp spaces.
The Lp spaces.
Equivalent norms.
Bounded operators and linear functionals.
Inner product spaces.
Closest points and the projection theorem.
Orthonormal bases and Parseval's equation.
Orthonormal bases and the Gram-Schmidt process.
The dual of a Hilbert space: the Riesz representation theorem.
Fourier series and the Dirichlet problem.
The Dirichlet problem.
Uniformly convergent Fourier series.
Application to theta, gamma and zeta functions.
Theta functions.
The gamma and beta functions.
The Riemann zeta function.