Functional analysis is a branch of mathematical analysis. It is used to explain the workings of a complex system. It was formed by the study of vector spaces with the limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense. Fourier transform as transformations defining continuous, unitary operators between function spaces. This analysis is useful for the study of differential and integral equations.

The basic idea of functional analysis is that the system is viewed as computing a function. It assumes that such processing can be explained by decomposing this complex function into a set of simpler functions that are computed by an organized system of sub processors. There are three stages in the methodology that defines functional analysis. The first stage is where the to-be-explained function is defined. The second stage is where the analysis is performed. The to-be-explained function is decomposed into an organized set of simpler functions. This analysis can precede recursively by decomposing some of the subfunctions into sub-subfunctions. The third stage is where the analysis is stopped by subsuming the bottom level of functions. The operation of each of these operations is explained by appealing to natural laws.

In modern mathematics, functional analysis is the subject seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces. An important aspect 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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