What is "functional analysis"

n. (context mathematics English) The branch of mathematics dealing with infinite-dimensional vector spaces, whose elements are actually functions, as well as generalizations such as Banach spaces and Hilbert spaces.

Functional analysis is a branch of mathematical 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 functions defined on these spaces and respecting these structures in a suitable sense. 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 a 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, and 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.

Functional analysis in behavioral psychology is the application of the laws of operant conditioning to establish the relationships between stimuli and responses. To establish the function of a behavior, one typically examines the "four-term contingency": first by identifying the motivating operations (EO or AO), then identifying the antecedent or trigger of the behavior, identifying the behavior itself as it has been operationalized, and identifying the consequence of the behavior which continues to maintain it.

Functional assessment in behavior analysis employs principles derived from the natural science of behavior analysis to determine the "reason", purpose, or motivation for a behavior. The most robust form of functional assessment is functional analysis, which involves the direct manipulation, using some experimental design (e.g., a multielement design or a reversal design) of various antecedent and consequent events and measurement of their effects on the behavior of interest; this is the only method of functional assessment that allows for demonstration of clear cause of behavior.