Quasi-derivative

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 Gâteaux 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 "x"₀ ∈ "A" is a linear transformation "u" : "E" → "F" with the following property: for every continuous function "g" : [0,1] → "A" with "g"(0)="x"₀ such that "g"′(0) ∈ "E" exists,

:$lim\_\{t\; o\; 0^+\}frac\{f(g(t))-f(x\_0)\}\{t\}\; =\; u(g\text{'}(0)).$

If such a linear map "u" exists, then "f" is said to be "quasi-differentiable" at "x"₀.

Continuity of "u" need not be assumed, but it follows instead from the definition of the quasi-derivative. If "f" is Fréchet differentiable at "x"₀, then by the chain rule, "f" is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at "x"₀. The converse is true provided "E" is finite dimensional. Finally, if "f" is quasi-differentiable, then it is Gâteaux differentiable and its Gâteaux derivative is equal to its quasi-derivative.

References

*cite book|author=Dieudonné, J|title=Foundations of modern analysis|publisher=Academic Press|year=1969