FUNCTIONAL ANALYSIS is the branch of mathematics concerned with the modern abstract study of linear and non-linear functions in terms of the underlying linear (topological, normed, Banach, Hilbert, ...) spaces on which the functions are defined and the duals of those spaces. This perspective, growing out of the study of linear operators and functionals, aims at producing a unifying corpus of results and techniques for linear spaces and linear operators. This is applicable to the study of such diverse areas of mathematics as algebra, real analysis, calculus of variations, numerical analysis, differential equations, and differential geometry, through the application of general theorems such as the Hahn-Banach Theorem, the Uniform Boundedness Principle, the Open Mapping Theorem, and the Riesz Representation Theorem. (Twisted somewhat from the Harper Collins Mathematics Dictionary.)