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.