Functional analysis is a branch of mathematics and a generalization of classical analysis. It integrates the viewpoints of function theory, geometry and algebra to study functions, operators and limit theory in infinite vector space. Functional analysis is widely used in physics, engineering, computer science and other fields.

For example, the most important application of linear functional analysis is to lay a theoretical foundation for linear integral equations and linear partial differential equations. The most important applications of nonlinear functional analysis are nonlinear dynamical systems, nonlinear partial differential equations (PDE) and variational methods (called optimal control in engineering or economics).