In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the quasi-derivative is one of several generalizations of the
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of a
function between two
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) 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 vectors and ...
s. 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
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
from an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
''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,
:
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 ...
, ''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