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{{about|an area of mathematics|a method of study of human behavior|Functional analysis (psychology)|a method in linguistics|Functional analysis (linguistics)}}
[[Image:Drum vibration mode12.gif|thumb|right|200px|link=https://en.m.wikipedia.org/wiki/Vibrations_of_a_circular_membrane |One of the possible modes of vibration of an idealized circular [[drum head]]. These modes are [[eigenfunction]]s of a linear operator on a function space, a common construction in functional analysis.]]
'''Functional analysis''' is a branch of [[mathematical analysis]], the core of which is formed by the study of [[vector space]]s endowed with some kind of limit-related structure (for example, [[Inner product space#Definition|inner product]], [[Norm (mathematics)#Definition|norm]], or [[Topological space#Definition|topology]]) and the [[linear transformation|linear function]]s defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of [[function space|spaces of functions]] and the formulation of properties of transformations of functions such as the [[Fourier transform]] as transformations defining, for example, [[continuous function|continuous]] or [[unitary operator|unitary]] operators between function spaces. This point of view turned out to be particularly useful for the study of [[differential equations|differential]] and [[integral equations]].
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