Gâteaux derivative, weak derivativeThe derivative of a functional or a mapping which — together with the
Fréchet derivative
(the
strong derivative)
— is most frequently
used in infinite-dimensional analysis. The
Gâteaux derivative
at a point
of a mapping
from a linear topological space
into a linear topological space
is the continuous linear mapping
that satisfies the condition
where
as
in the topology of
(see also
Gâteaux variation).
If the mapping
has a Gâteaux derivative at the point
 ,
it is called
Gâteaux differentiable.
The theorem on differentiation of a composite function is
usually invalid for the Gâteaux derivative. See also
Differentiation of a mapping.
References| [1] |
R. Gâteaux,
"Sur les fonctionnelles continues et les fonctionnelles analytiques"
C.R. Acad. Sci. Paris Sér. I Math.
, 157
(1913)
pp. 325–327 | | [2] |
A.N. Kolmogorov,
S.V. Fomin,
"Elements of the theory of functions and functional analysis"
, 1–2
, Graylock
(1957–1961)
(Translated from Russian) | | [3] |
W.I. [V.I. Sobolev] Sobolew,
"Elemente der Funktionalanalysis"
, H. Deutsch
, Frankfurt a.M.
(1979)
(Translated from Russian) | | [4] |
V.I. Averbukh,
O.G. Smolyanov,
"Theory of differentiation in linear topological spaces"
Russian Math. Surveys
, 22
: 6
(1967)
pp. 201–258
Uspekhi Mat. Nauk
, 22
: 6
(1967)
pp. 201–260 |
V.M. Tikhomirov
CommentsReferences| [a1] |
M.S. Berger,
"Nonlinearity and functional analysis"
, Acad. Press
(1977) |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|