- Gâteaux derivative
In
mathematics , the Gâteaux differential is a generalisation of the concept ofdirectional derivative indifferential calculus . Named afterRené Gâteaux , a French mathematician who died young inWorld War I , it is defined for functions betweenlocally convex topological vector space s such asBanach space s. Like theFréchet derivative on a Banach space, the Gâteaux differential is often used to formalize thefunctional derivative commonly used in thecalculus of variations andphysics . Unlike other forms of derivatives, the Gâteaux differential of a function may benonlinear . If the Gâteaux differential is linear and continuous, then the resulting linear operator is called the Gâteaux derivative.Definition
Suppose and are
locally convex topological vector space s (for example,Banach space s), is open, and:
The Gâteaux differential of at in the direction is defined as
:
if the limit exists. If the limit exists for all , then one says that has Gâteaux differential at . If "X" and "Y" are complex topological vector spaces, then the limit above is usually taken as τ→0 in the
complex plane . If, on the other hand, "X" and "Y" are real, then the limit is taken for real τ.The Gâteaux differential may fail to be linear or, being linear, it may fail to be a
continuous linear transformation . If the mapping:
is continuous and linear, then it is called the Gâteaux derivative of "F" at "u" and "F" is said to be Gâteaux differentiable at "u".
Continuous Gâteaux differentiability may be defined in two inequivalent ways. Suppose that "F":"U"→"Y" is Gâteaux differentiable at each point of the open set "U". One notion of continuous differentiability in "U" requires that the mapping on the
product space :
be continuous. Another notion requires continuity of the mapping
:
be a continuous mapping
:
from "U" to the space of continuous linear functions from "X" to "Y". The latter notion of continuous differentiability is typical (but not universal) when the spaces "X" and "Y" are Banach. The former is the more common definition in applications such as the
Nash-Moser inverse function theorem .Properties
If the Gâteaux differential exists, it is unique.
For each the Gâteaux differential is an operator
:
This operator is homogeneous, so that
:,
but it is not additive in general case, and, hence, is not always linear, unlike the
Fréchet derivative .However, suppose that "X" and "Y" complex Banach spaces, "u" is a point of the open set "U"⊂"X", and "F":"U" → "Y". Then if "F" is (complex) Gâteaux differentiable at each "u" ∈ "U" with derivative
:
then "F" is Fréchet differentiable on "U" with Fréchet derivative "DF" harv|Zorn|1946. This is analogous to the result from basic
complex analysis that a function isanalytic if it is complex differentiable, and is a fundamental result in the study ofinfinite dimensional holomorphy .If "F" is Fréchet differentiable, then it is also Gâteaux differentiable, and its Fréchet and Gâteaux derivatives agree.
Example
Let be the
Hilbert space ofsquare-integrable function s on a Lebesgue measurable set in theEuclidean space R"N". The functional:
given by
:
where "F" is a real-valued function of a real variable with "F"′ = ƒ and "u" is defined on Ω with real values, has Gâteaux derivative :
Indeed,
:::
Letting τ → 0 gives the Gâteaux derivative:that is, the inner product 〈ƒ,ψ〉.
See also
*
Derivative (generalizations)
*Differentiation in Fréchet spaces References
* citation | first = R|last=Gâteaux | title =Sur les fonctionnelles continues et les fonctionnelles analytiques | pages = | url = http://gallica.bnf.fr/ | journal = Comptes rendus de l'academie des sciences|publication-place=Paris|volume=157|year=1913 | pages = 325-327 | accessmonthday=30 July |accessyear = 2006 .
*.
*.
*|doi=10.1090/S0002-9904-1946-08524-9.
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