In mathematics, functional analysis refers to studies where the essence is the use of abstract methods to study problems where sets of numerical functions are involved. These "abstract methods" consist of applying theorems about functions between sets having an algebraic and a topological or a more general limit structure, i.e. one defining some kind of "limit process". One generally calls such structured sets spaces, and the functions between them are called maps. Linear functional analysis refers to the part of the discipline where only linear maps are involved. In nonlinear functional analysis, one considers also nonlinear maps. An example of the application of functional analytic methods would be the use of a fixed point theorem to show existence of a solution to a differential equation.
Functional analysis is a branch of analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit related structure (e.g. inner product, norm, topology, etc.) and the linear maps acting upon these spaces and respecting these structures in a suitable sense. One generally calls these linear maps operators. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that subject. However, the general concept of functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces. In contrast, linear algebra deals mostly with finite dimensional spaces, or does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.
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