Fréchet derivative

The Fréchet derivative of a function between normed spaces is a bounded linear map that approximates the function’s change to first order, with an error small relative to the input change.

Fréchet differentiability: A function is Fréchet differentiable at a point when its change equals a bounded linear approximation plus an error negligible relative to the input change. This property is defined by the existence of the derivative at the point.
Normed vector space: A vector space equipped with a norm that assigns lengths to vectors. Fréchet derivatives compare input and output errors using norms.
Gâteaux derivative: The Gâteaux derivative is a directional linear approximation defined by taking a limit along each fixed direction. It tests directions separately, unlike the uniform small-error requirement of Fréchet differentiability.
Newton's method in Banach spaces: Newton's method in Banach spaces iteratively solves nonlinear equations using linearizations by Fréchet derivatives. Each step uses the derivative to construct a linear correction to the current estimate.
Chain rule: The chain rule gives the derivative of a composition in terms of the derivatives of its component functions. For Fréchet derivatives, composition corresponds to composing the associated linear maps.
Fréchet derivative definition: The limit definition requires the approximation error divided by the input-change norm to tend to zero. This is the precise condition the derivative must satisfy.
Linear map: A function between vector spaces that preserves addition and scalar multiplication. The first-order approximation must be linear in the input change.
Hadamard derivative: The Hadamard derivative strengthens directional differentiation by requiring stability as the direction varies with the increment. It lies between directional and Fréchet differentiability in common settings.
Implicit function theorem: The implicit function theorem gives conditions for solving an equation locally for some variables as functions of others. Its Banach-space version uses an invertible partial Fréchet derivative.
Taylor's theorem: Taylor's theorem approximates a sufficiently smooth function by a polynomial and bounds the remainder. The Fréchet derivative supplies the first-order term in Taylor expansions on normed spaces.

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