Quasi-derivative

Inmathematics,thequasi-derivativeis one of several generalizations of thederivativeof afunctionbetween twoBanach spaces.The quasi-derivative is a slightly stronger version of theGateaux derivative,though weaker than theFréchet derivative.

Letf:A→Fbe acontinuous functionfrom anopen setAin a Banach spaceEto another Banach spaceF.Then thequasi-derivativeoffatx₀∈Ais alinear transformationu:E→Fwith the following property: for every continuous functiong:[0,1] →Awithg(0)=x₀such thatg′(0) ∈Eexists,

\lim _{t\to 0^{+}}{\frac {f(g(t))-f(x_{0})}{t}}=u(g'(0)).

If such a linear mapuexists, thenfis said to bequasi-differentiableatx₀.

Continuity ofuneed not be assumed, but it follows instead from the definition of the quasi-derivative. Iffis Fréchet differentiable atx₀,then by thechain rule,fis also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative atx₀.The converse is true providedEis finite-dimensional. Finally, iffis 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.

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