Functional analysis and the calculus of variations deals with the real analysis formalism for functionals and operators defined on spaces of infinite dimension. In particular, the calculus of variations generalizes and extends the standard differential and integral calculus theory for n real variables to infinite dimensional spaces of real functions. The book discusses the basic concepts of functional analysis, measure and integration theory, calculus of variations and duality aiming applications to variational problems of non-convex nature, such as the Ginzburg-Landau system in superconductivity, shape optimization models, dual variational formulations for micro-magnetism and others.