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A mapping $f$ of an arbitrary set $X$ into the set $\mathbb R$ of real numbers or the set $\mathbb C$ of complex numbers. If $X$ is endowed with the structure of a vector space, a topological space or an ordered set, then there arise the important classes of linear, continuous and monotone functionals, respectively (cf. Linear functional; Continuous functional; Monotone mapping).


[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
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Functional. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article