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}}

Latest revision as of 01:04, 4 January 2015

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