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Difference between revisions of "Gâteaux differential"

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m (moved Gateaux differential to Gâteaux differential over redirect: accented title)
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''of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g0433701.png" /> of a linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g0433702.png" /> into a linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g0433703.png" />''
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''of a mapping  $  f $
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of a linear topological space $  X $
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into a linear topological space $  Y $''
  
 
The function
 
The function
  
<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/g/g043/g043370/g0433704.png" /></td> </tr></table>
+
$$
 +
h  \rightarrow  Df ( x _ {0} , h),
 +
$$
  
 
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/g/g043/g043370/g0433705.png" /></td> </tr></table>
+
$$
 
+
Df ( x _ {0} , h)  =  \left .
<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/g/g043/g043370/g0433706.png" /></td> </tr></table>
 
 
 
on the assumption that the limit exists for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g0433707.png" />, the convergence being understood in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g0433708.png" />. The Gâteaux differential thus defined is homogeneous, but is not additive. Gâteaux differentials of higher orders are defined in a similar manner. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g0433709.png" /> is sometimes known as the [[Gâteaux variation|Gâteaux variation]] or as the weak differential. See also [[Differentiation of a mapping|Differentiation of a mapping]]; [[Variation|Variation]].
 
 
 
Linearity and continuity are usually additionally stipulated: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337011.png" />. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337012.png" /> is known as the [[Gâteaux derivative|Gâteaux derivative]]. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337013.png" /> is uniformly continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337014.png" /> and continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337015.png" /> in some domain, then the [[Fréchet derivative|Fréchet derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337017.png" /> exists in this domain and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043370/g04337018.png" />.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.I. [V.I. Sobolev] Sobolew,   "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M.  (1979(Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>
 
  
 +
\frac{d}{dt }
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f
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( x _ {0} + th) \right | _ {t = 0 }  =
 +
$$
  
 +
$$
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= \
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\lim\limits _ {t \rightarrow 0 } 
 +
\frac{f ( x _ {0} + th) - f ( x _ {0} ) }{t}
 +
,
 +
$$
  
====Comments====
+
on the assumption that the limit exists for all  $  h \in X $,
 +
the convergence being understood in the topology of  $  Y $.
 +
The Gâteaux differential thus defined is homogeneous, but is not additive. Gâteaux differentials of higher orders are defined in a similar manner. The mapping  $  h \rightarrow Df( x _ {0} , h) $
 +
is sometimes known as the [[Gâteaux variation|Gâteaux variation]] or as the weak differential. See also [[Differentiation of a mapping|Differentiation of a mapping]]; [[Variation|Variation]].
  
 +
Linearity and continuity are usually additionally stipulated:  $  Df( x, h) = f _ {G} ^ { \prime } ( x _ {0} ) h $,
 +
$  f _ {G} ^ { \prime } ( x _ {0} ) \in L( X, Y) $.
 +
In such a case  $  f _ {G} ^ { \prime } ( x _ {0} ) $
 +
is known as the [[Gâteaux derivative|Gâteaux derivative]]. If the mapping  $  ( x, h) \rightarrow Df( x, h) $
 +
is uniformly continuous in  $  x $
 +
and continuous in  $  h $
 +
in some domain, then the [[Fréchet derivative|Fréchet derivative]]  $  f ^ { \prime } $
 +
of  $  f $
 +
exists in this domain and  $  f ^ { \prime } ( x) h = Df ( x, h ) $.
  
 
====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">[1]</TD> <TD valign="top">  W.I. [V.I. Sobolev] Sobolew,  "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M.  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Berger,  "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR>
 +
</table>

Latest revision as of 14:29, 15 April 2023


of a mapping $ f $ of a linear topological space $ X $ into a linear topological space $ Y $

The function

$$ h \rightarrow Df ( x _ {0} , h), $$

where

$$ Df ( x _ {0} , h) = \left . \frac{d}{dt } f ( x _ {0} + th) \right | _ {t = 0 } = $$

$$ = \ \lim\limits _ {t \rightarrow 0 } \frac{f ( x _ {0} + th) - f ( x _ {0} ) }{t} , $$

on the assumption that the limit exists for all $ h \in X $, the convergence being understood in the topology of $ Y $. The Gâteaux differential thus defined is homogeneous, but is not additive. Gâteaux differentials of higher orders are defined in a similar manner. The mapping $ h \rightarrow Df( x _ {0} , h) $ is sometimes known as the Gâteaux variation or as the weak differential. See also Differentiation of a mapping; Variation.

Linearity and continuity are usually additionally stipulated: $ Df( x, h) = f _ {G} ^ { \prime } ( x _ {0} ) h $, $ f _ {G} ^ { \prime } ( x _ {0} ) \in L( X, Y) $. In such a case $ f _ {G} ^ { \prime } ( x _ {0} ) $ is known as the Gâteaux derivative. If the mapping $ ( x, h) \rightarrow Df( x, h) $ is uniformly continuous in $ x $ and continuous in $ h $ in some domain, then the Fréchet derivative $ f ^ { \prime } $ of $ f $ exists in this domain and $ f ^ { \prime } ( x) h = Df ( x, h ) $.

References

[1] W.I. [V.I. Sobolev] Sobolew, "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M. (1979) (Translated from Russian)
[2] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[a1] M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)
How to Cite This Entry:
Gâteaux differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_differential&oldid=23302
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article