In mathematics, the quasi-derivative is one of several generalizations of the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

between two

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

s. The quasi-derivative is a slightly stronger version of the

Gateaux derivative In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...

, though weaker than the

Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-value ...

. Let ''f'' : ''A'' → ''F'' be a continuous function from an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...

''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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

''u'' : ''E'' → ''F'' with the following property: for every continuous function ''g'' : ,1→ ''A'' with ''g''(0)=''x''₀ such that ''g''′(0) ∈ ''E'' exists, :

\lim_\frac = u(g'(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 In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...

, ''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 Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.

References

* {{mathanalysis-stub Banach spaces Generalizations of the derivative