To better prepare students to learn the variational theory of partial differential equations and numerical analysis, this textbook presents mathematical foundations leading to classical results in functional analysis. Significantly revised and expanded, this second edition provides new examples, new exercises, and a new solutions manual for qualifying instructors. Each chapter in this edition features an extensive introduction, a summary, and historical comments. Additional subjects addressed in the text include singular value decomposition, the Lebesgue measure, the Banach contractive map theorem, Schwartz distributions, and elementary spectral theory.