The derivative of a functional or a mapping which — together with the Fréchet derivative (the strong derivative) — is most frequently used in infinite-dimensional analysis. The Gâteaux derivative at a point of a mapping from a linear topological space into a linear topological space is the continuous linear mapping that satisfies the condition
where as in the topology of (see also Gâteaux variation). If the mapping has a Gâteaux derivative at the point , it is called Gâteaux differentiable. The theorem on differentiation of a composite function is usually invalid for the Gâteaux derivative. See also Differentiation of a mapping.
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Gâteaux derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_derivative&oldid=13686