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The basic and historically first class of spaces studied in functional analysis are [[complete space|complete]] [[normed vector space]]s over the [[real number|real]] or [[complex number]]s. Such spaces are called [[Banach space]]s. An important example is a [[Hilbert space]], where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the [[mathematical formulation of quantum mechanics]], [[Reproducing kernel Hilbert space|machine learning]], [[partial differential equations]], and [[Fourier analysis]].{{cn}}
More generally, functional analysis includes the study of [[Fréchet space]]s and other [[topological vector space]]s not endowed with a norm.{{cn}}
An important object of study in functional analysis are the [[continuous function (topology)|continuous]] [[linear transformation|linear operators]] defined on Banach and Hilbert spaces. These lead naturally to the definition of [[C*-algebra]]s and other [[operator algebra]]s.
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