Let f : A → F be a continuous function from an open setA in a Banach space E to another Banach space F. Then the quasi-derivative of f at x0 ∈ A is a linear transformationu : E → F with the following property: for every continuous function g : [0,1] → A with g(0)=x0 such that g′(0) ∈ E exists,
If such a linear map u exists, then f is said to be quasi-differentiable at x0.
Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x0. The converse is true provided E is finite-dimensional. Finally, if f is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.
Referencesedit
Dieudonné, J (1969). Foundations of modern analysis. Academic Press.
quasi, derivative, mathematics, quasi, derivative, several, generalizations, derivative, function, between, banach, spaces, quasi, derivative, slightly, stronger, version, gateaux, derivative, though, weaker, than, fréchet, derivative, continuous, function, fr. In mathematics the quasi derivative is one of several generalizations of the derivative of a function between two Banach spaces The quasi derivative is a slightly stronger version of the Gateaux derivative though weaker than the Frechet derivative Let f A F be a continuous function from an open set A in a Banach space E to another Banach space F Then the quasi derivative of f at x0 A is a linear transformation u E F with the following property for every continuous function g 0 1 A with g 0 x0 such that g 0 E exists lim t 0 f g t f x 0 t u g 0 displaystyle lim t to 0 frac f g t f x 0 t u g 0 If such a linear map u exists then f is said to be quasi differentiable at x0 Continuity of u need not be assumed but it follows instead from the definition of the quasi derivative If f is Frechet differentiable at x0 then by the chain rule f is also quasi differentiable and its quasi derivative is equal to its Frechet derivative at x0 The converse is true provided E is finite dimensional Finally if f is quasi differentiable then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi derivative References editDieudonne J 1969 Foundations of modern analysis Academic Press nbsp This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Quasi derivative amp oldid 1119695882, wikipedia, wiki, book, books, library,
