List of Indiana Pacers seasons: Difference between revisions

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In [[mathematics]], the '''quasi-derivative''' is one of several generalizations of the [[derivative]] of a [[function (mathematics)|function]] between two [[Banach space]]s. The quasi-derivative is a slightly stronger version of the [[Gâteaux derivative]], though weaker than the [[Fréchet derivative]].


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Let ''f'' : ''A'' &rarr; ''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''<sub>0</sub> &isin; ''A'' is a [[linear transformation]] ''u'' : ''E'' &rarr; ''F'' with the following property: for every continuous function ''g'' : [0,1] &rarr; ''A'' with ''g''(0)=''x''<sub>0</sub> such that ''g''&prime;(0) &isin; ''E'' exists,
 
:<math>\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 ''x''<sub>0</sub>.
 
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''<sub>0</sub>, then by the [[chain rule]], ''f'' is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at ''x''<sub>0</sub>. 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}}
 
[[Category:Banach spaces]]
[[Category:Generalizations of the derivative]]
 
 
 
{{mathanalysis-stub}}

Revision as of 12:18, 26 December 2013

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 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,

limt→0+f(g(t))−f(x0)t=u(g′(0)).

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 Gâteaux differentiable and its Gâteaux derivative is equal to its quasi-derivative.

References

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