This book provides the reader with a systematic exposition of the basic ideas and results of Hilbert space theory and functional analysis with diverse applications to differential and integral equations. The Hilbert space formalism is used to develop the foundation of quantum mechanics and the Hilbert space methods are applied to optimization, variational, and control problems and to problems in approximation theory, nonlinear instablity, and bifurcation. Another attractive feature is a simple introduction to the Lebesgue integral. It is intended for senior undergraduate and graduate courses in Hilbert space and functional analysis with applications for students in mathematics, physics, and engineering. n Systematic exposition of the basic ideas and results of Hilbert space theory and functional analysis n Great variety of applications that are not available in comparable books n Different approach to the Lebesgue integral, which makes the theory easier, more intuitive, and more accessible to undergraduate students