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''of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g0433901.png" /> of a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g0433902.png" /> into a linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g0433903.png" />''
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The limit, in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g0433904.png" />,
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<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/g043390/g0433905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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''of a mapping  $  f $
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of a linear space  $  X $
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into a linear topological space  $  Y $''
  
<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/g043390/g0433906.png" /></td> </tr></table>
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The limit, in the topology of  $  Y $,
  
on the assumption that it exists for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g0433907.png" />. This is how the first variation was introduced by R. Gâteaux in 1913–1914. This definition for functionals of the classical calculus of variations was given by J.L. Lagrange (cf. [[Variation of a functional|Variation of a functional]]).
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$$ \tag{* }
 +
\delta f ( x _ {0} , h)  = \left .
 +
{
 +
\frac{d}{dt }
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} f ( x _ {0} + th)
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\right | _ {t = 0 }  =
 +
$$
  
The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g0433908.png" /> need not necessarily be a linear functional in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g0433909.png" />, but it is always a homogeneous function of the first degree in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g04339010.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g04339011.png" /> is also known as the [[Gâteaux differential|Gâteaux differential]] or weak differential. Beginning with the work of P. Lévy, it is usual to stipulate the linearity and continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g04339012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g04339013.png" />:
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$$
 +
= \
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\lim\limits _ {t \rightarrow 0
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\frac{f ( x _ {0} + th) - f ( x _ {0} ) }{t }
  
<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/g043390/g04339014.png" /></td> </tr></table>
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$$
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043390/g04339015.png" /> is called the [[Gâteaux derivative|Gâteaux derivative]]. Second, etc., variations are defined similarly to (*). See also [[Variation|Variation]]; [[Second variation|Second variation]]; [[Differentiation of a mapping|Differentiation of a mapping]].
 
 
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  R. Gateaux,  "Sur les fonctionnelles continues et les fonctionnelles analytiques"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''157'''  (1913)  pp. 325–327</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  R. Gateaux,  "Fonctions d'une infinités des variables indépendantes"  ''Bull. Soc. Math. France'' , '''47'''  (1919)  pp. 70–96</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Lévy,  "Leçons d'analyse fonctionnelle" , Gauthier-Villars  (1922)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Lévy,  "Problèmes concrets d'analyse fonctionelle" , Gauthier-Villars  (1951)</TD></TR></table>
 
  
 +
on the assumption that it exists for all  $  h \in X $.
 +
This is how the first variation was introduced by R. Gâteaux in 1913–1914. This definition for functionals of the classical calculus of variations was given by J.L. Lagrange (cf. [[Variation of a functional|Variation of a functional]]).
  
 +
The expression  $  \delta f( x _ {0} , h) $
 +
need not necessarily be a linear functional in  $  h $,
 +
but it is always a homogeneous function of the first degree in  $  h $.
 +
The mapping  $  h \rightarrow \delta f ( x _ {0} , h) $
 +
is also known as the [[Gâteaux differential|Gâteaux differential]] or weak differential. Beginning with the work of P. Lévy, it is usual to stipulate the linearity and continuity of  $  \delta f ( x _ {0} , h) $
 +
in  $  h $:
  
====Comments====
+
$$
 +
\delta f ( x _ {0} , h)  = f _ {G} ^ { \prime } ( x _ {0} ) h,\ \
 +
f _ {G} ^ { \prime } ( x _ {0} )  \in  L ( X, Y).
 +
$$
  
 +
where  $  f _ {G} ^ { \prime } ( x _ {0} ) $
 +
is called the [[Gâteaux derivative|Gâteaux derivative]]. Second, etc., variations are defined similarly to (*). See also [[Variation|Variation]]; [[Second variation|Second variation]]; [[Differentiation of a mapping|Differentiation of a mapping]].
  
 
====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">[1a]</TD> <TD valign="top">  R. Gateaux,  "Sur les fonctionnelles continues et les fonctionnelles analytiques"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''157'''  (1913)  pp. 325–327</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  R. Gateaux,  "Fonctions d'une infinités des variables indépendantes"  ''Bull. Soc. Math. France'' , '''47'''  (1919)  pp. 70–96</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Lévy,  "Leçons d'analyse fonctionnelle" , Gauthier-Villars  (1922)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Lévy,  "Problèmes concrets d'analyse fonctionelle" , Gauthier-Villars  (1951)</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 space $ X $ into a linear topological space $ Y $

The limit, in the topology of $ Y $,

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

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

on the assumption that it exists for all $ h \in X $. This is how the first variation was introduced by R. Gâteaux in 1913–1914. This definition for functionals of the classical calculus of variations was given by J.L. Lagrange (cf. Variation of a functional).

The expression $ \delta f( x _ {0} , h) $ need not necessarily be a linear functional in $ h $, but it is always a homogeneous function of the first degree in $ h $. The mapping $ h \rightarrow \delta f ( x _ {0} , h) $ is also known as the Gâteaux differential or weak differential. Beginning with the work of P. Lévy, it is usual to stipulate the linearity and continuity of $ \delta f ( x _ {0} , h) $ in $ h $:

$$ \delta f ( x _ {0} , h) = f _ {G} ^ { \prime } ( x _ {0} ) h,\ \ f _ {G} ^ { \prime } ( x _ {0} ) \in L ( X, Y). $$

where $ f _ {G} ^ { \prime } ( x _ {0} ) $ is called the Gâteaux derivative. Second, etc., variations are defined similarly to (*). See also Variation; Second variation; Differentiation of a mapping.

References

[1a] R. Gateaux, "Sur les fonctionnelles continues et les fonctionnelles analytiques" C.R. Acad. Sci. Paris Sér. I Math. , 157 (1913) pp. 325–327
[1b] R. Gateaux, "Fonctions d'une infinités des variables indépendantes" Bull. Soc. Math. France , 47 (1919) pp. 70–96
[2] P. Lévy, "Leçons d'analyse fonctionnelle" , Gauthier-Villars (1922)
[3] P. Lévy, "Problèmes concrets d'analyse fonctionelle" , Gauthier-Villars (1951)
[a1] M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)
How to Cite This Entry:
Gâteaux variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_variation&oldid=14647
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article