This second edition is thoroughly revised and includes several new examples and exercises. Proofs of many results have been rewritten for a greater clarity. While covering all the standard material expected of such a course, efforts have been made to illustrate the use of the topics to study differential equations and calculus of variations. The book includes a chapter on weak topologies and their applications. It also includes a chapter on the Lebesgue spaces, which discusses Sobolev spaces. The book includes a chapter on compact operators and their spectra, especially for compact self-adjoint operators on a Hilbert space. Each chapter has a large collection of exercises in the end, which give additional examples and counterexamples to the results given in the text. This book is suitable for a first course in functional analysis for graduate students who wish to pursue a career in the applications of mathematics.