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Concept
Fréchet derivative
Summary
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations.
Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces. The Fréchet derivative should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative.
The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis.
Let and be normed vector spaces, and be an open subset of A function is called Fréchet differentiable at if there exists a bounded linear operator such that
The limit here is meant in the usual sense of a limit of a function defined on a metric space (see Functions on metric spaces), using and as the two metric spaces, and the above expression as the function of argument in As a consequence, it must exist for all sequences of non-zero elements of that converge to the zero vector Equivalently, the first-order expansion holds, in Landau notation
If there exists such an operator it is unique, so we write and call it the Fréchet derivative of at
A function that is Fréchet differentiable for any point of is said to be C1 if the function
is continuous ( denotes the space of all bounded linear operators from to ). Note that this is not the same as requiring that the map be continuous for each value of (which is assumed; bounded and continuous are equivalent).
This notion of derivative is a generalization of the ordinary derivative of a function on the real numbers since the linear maps from to are just multiplication by a real number.
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