Functional analysis is a subfield of mathematics which studies problems in analysis, including differential equations, integral equations, Brownian motion and probability, by thinking of the functions in question as elements of vector spaces, and studying the behavior of the linear mappings or operators which connect them. In this sense functional analysis is an infinite-dimensional generalization of linear algebra.

In a more subtle sense, functional analysis can be thought of as a generalization of calculus in which multiplication is not always commutative, since number-valued functions are replaced by operator-valued ones. In this form it is needed to state the theory of quantum physics, since at the quantum level, the order in which measurements are performed can affect the results.

Objects and ideas connected with functional analysis include Banach spaces, Hilbert spaces, linear operators, spectra, and the Lebesgue integral.

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