Quasi-derivative

Short description: Generalization of a derivative of a function between two Banach spaces

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 Fréchet 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,

[math]\displaystyle{ \lim_{t\to 0^+}\frac{f(g(t))-f(x_0)}{t} = u(g'(0)). }[/math]

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.

References

• Dieudonné, J (1969). Foundations of modern analysis. Academic Press.