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''strong derivative''
 
''strong derivative''
  
The most widespread (together with the [[Gâteaux derivative|Gâteaux derivative]], which is sometimes called the weak derivative) derivative of a functional or a mapping. The Fréchet derivative of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413601.png" /> of a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413602.png" /> into a normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413603.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413604.png" /> is the linear continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413605.png" /> satisfying the condition
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The most widespread (together with the [[Gâteaux derivative|Gâteaux derivative]], which is sometimes called the weak derivative) derivative of a functional or a mapping. The Fréchet derivative of a mapping $  f: X \rightarrow Y $
 +
of a normed space $  X $
 +
into a normed space $  Y $
 +
at a point $  x _ {0} $
 +
is the linear continuous operator $  \Lambda : X \rightarrow Y $
 +
satisfying the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413606.png" /></td> </tr></table>
+
$$
 +
f ( x _ {0} + h)  = \
 +
f ( x _ {0} ) + \Lambda h + \epsilon ( h),
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413607.png" /></td> </tr></table>
+
$$
 
+
\lim\limits _ {\| h \| \rightarrow 0 } \
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413608.png" /> satisfying these conditions is unique (if it exists) and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f0413609.png" />; the linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136010.png" /> is called the [[Fréchet differential|Fréchet differential]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136011.png" /> has a Fréchet derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136012.png" />, it is said to be Fréchet differentiable. The most important theorems of differential calculus hold for Fréchet derivatives — the theorem on the differentiation of a composite function and the mean value theorem. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136013.png" /> is continuously Fréchet differentiable in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136014.png" /> and if the Fréchet derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136016.png" /> is a homeomorphism of the Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041360/f04136018.png" />, then the inverse mapping theorem holds. See also [[Differentiation of a mapping|Differentiation of a mapping]].
 
  
 +
\frac{\| \epsilon ( h) \| }{\| h \| }
 +
  =  0.
 +
$$
  
 +
The operator  $  \Lambda $
 +
satisfying these conditions is unique (if it exists) and is denoted by  $  f ^ { \prime } ( x _ {0} ) $;
 +
the linear mapping  $  h \rightarrow f ^ { \prime } ( x _ {0} ) h $
 +
is called the [[Fréchet differential|Fréchet differential]]. If  $  f $
 +
has a Fréchet derivative at  $  x _ {0} $,
 +
it is said to be Fréchet differentiable. The most important theorems of differential calculus hold for Fréchet derivatives — the theorem on the differentiation of a composite function and the mean value theorem. If  $  f $
 +
is continuously Fréchet differentiable in a neighbourhood of a point  $  x _ {0} $
 +
and if the Fréchet derivative  $  f ^ { \prime } ( x _ {0} ) $
 +
at  $  x _ {0} $
 +
is a homeomorphism of the Banach spaces  $  X $
 +
and  $  Y $,
 +
then the inverse mapping theorem holds. See also [[Differentiation of a mapping|Differentiation of a mapping]].
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Berger,  "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Berger,  "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR></table>

Revision as of 19:40, 5 June 2020


strong derivative

The most widespread (together with the Gâteaux derivative, which is sometimes called the weak derivative) derivative of a functional or a mapping. The Fréchet derivative of a mapping $ f: X \rightarrow Y $ of a normed space $ X $ into a normed space $ Y $ at a point $ x _ {0} $ is the linear continuous operator $ \Lambda : X \rightarrow Y $ satisfying the condition

$$ f ( x _ {0} + h) = \ f ( x _ {0} ) + \Lambda h + \epsilon ( h), $$

where

$$ \lim\limits _ {\| h \| \rightarrow 0 } \ \frac{\| \epsilon ( h) \| }{\| h \| } = 0. $$

The operator $ \Lambda $ satisfying these conditions is unique (if it exists) and is denoted by $ f ^ { \prime } ( x _ {0} ) $; the linear mapping $ h \rightarrow f ^ { \prime } ( x _ {0} ) h $ is called the Fréchet differential. If $ f $ has a Fréchet derivative at $ x _ {0} $, it is said to be Fréchet differentiable. The most important theorems of differential calculus hold for Fréchet derivatives — the theorem on the differentiation of a composite function and the mean value theorem. If $ f $ is continuously Fréchet differentiable in a neighbourhood of a point $ x _ {0} $ and if the Fréchet derivative $ f ^ { \prime } ( x _ {0} ) $ at $ x _ {0} $ is a homeomorphism of the Banach spaces $ X $ and $ Y $, then the inverse mapping theorem holds. See also Differentiation of a mapping.

Comments

References

[a1] M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)
How to Cite This Entry:
Fréchet derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_derivative&oldid=22455
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article