Derivative defined on normed spaces
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Differential Definitions |
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In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations.
Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces. The Fréchet derivative should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative.
The Fréchet derivative has applications to nonlinear problems throughout mathematical analysis and physical sciences, particularly to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis.
Definition
Let
and
be normed vector spaces, and
be an open subset of
A function
is called Fréchet differentiable at
if there exists a bounded linear operator
such that
![{\displaystyle \lim _{\|h\|_{V}\to 0}{\frac {\|f(x+h)-f(x)-Ah\|_{W}}{\|h\|_{V}}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3afc750cff7a26ddabc67e870b98a49f4815833f)
The limit here is meant in the usual sense of a limit of a function defined on a metric space (see Functions on metric spaces), using
and
as the two metric spaces, and the above expression as the function of argument
in
As a consequence, it must exist for all sequences
of non-zero elements of
that converge to the zero vector
Equivalently, the first-order expansion holds, in Landau notation
![{\displaystyle f(x+h)=f(x)+Ah+o(h).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb0bc4b20062d2506c94fd0f997de3f815e42bdf)
If there exists such an operator
it is unique, so we write
and call it the Fréchet derivative of
at
A function
that is Fréchet differentiable for any point of
is said to be C1 if the function
![{\displaystyle Df:U\to B(V,W);x\mapsto Df(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e326abb3477d1de102483cd87af8d41f8d254429)
is continuous (
![{\displaystyle B(V,W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e863f2fe931bd6e89ff8a2dc2be4615b283291e6)
denotes the space of all bounded linear operators from
![{\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
to
![{\displaystyle W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
). Note that this is not the same as requiring that the map
![{\displaystyle Df(x):V\to W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d1dfce47a26e53981807b708b7db9b7b2d8dc2)
be continuous for each value of
![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
(which is assumed; bounded and continuous are equivalent).
This notion of derivative is a generalization of the ordinary derivative of a function on the real numbers
since the linear maps from
to
are just multiplication by a real number. In this case,
is the function
Properties
A function differentiable at a point is continuous at that point.
Differentiation is a linear operation in the following sense: if
and
are two maps
which are differentiable at
and
is a scalar (a real or complex number), then the Fréchet derivative obeys the following properties:
![{\displaystyle D(cf)(x)=cDf(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2548a403fc873845ee1f62abac41f57493572e8e)
![{\displaystyle D(f+g)(x)=Df(x)+Dg(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27ff700d0fa26146c573e6049f72b5daf88b5521)
The chain rule is also valid in this context: if
is differentiable at
and
is differentiable at
then the composition
is differentiable in
and the derivative is the composition of the derivatives:
![{\displaystyle D(g\circ f)(x)=Dg(f(x))\circ Df(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b22df9ff9a0baeb516e7ac4dd2003d9ec1a71927)
Finite dimensions
The Fréchet derivative in finite-dimensional spaces is the usual derivative. In particular, it is represented in coordinates by the Jacobian matrix.
Suppose that
is a map,
with
an open set. If
is Fréchet differentiable at a point
then its derivative is
![{\displaystyle {\begin{cases}Df(a):\mathbb {R} ^{n}\to \mathbb {R} ^{m}\\Df(a)(v)=J_{f}(a)v\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab4401de1ef98c5f300f0fd71363e61208600a5)
where
![{\displaystyle J_{f}(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d602622c85d01faa032fe267d343360ccbdf98fd)
denotes the Jacobian matrix of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
at
Furthermore, the partial derivatives of
are given by
![{\displaystyle {\frac {\partial f}{\partial x_{i}}}(a)=Df(a)(e_{i})=J_{f}(a)e_{i},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e377eafab604aec205522c6fa91c04206bc6207)
where
![{\displaystyle \left\{e_{i}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76fb2ada40d7a9f01175c2c0b6276b396ca9f08d)
is the canonical basis of
![{\displaystyle \mathbb {R} ^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3)
Since the derivative is a linear function, we have for all vectors
![{\displaystyle h\in \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37d414ebb4e128e8959b70eb7da09f3517b7f0bc)
that the
directional derivative of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
along
![{\displaystyle h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
is given by
![{\displaystyle Df(a)(h)=\sum _{i=1}^{n}h_{i}{\frac {\partial f}{\partial x_{i}}}(a).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b64911ac668b2b80273a99895d698f9690df572)
If all partial derivatives of
exist and are continuous, then
is Fréchet differentiable (and, in fact, C1). The converse is not true; the function
![{\displaystyle f(x,y)={\begin{cases}(x^{2}+y^{2})\sin \left((x^{2}+y^{2})^{-1/2}\right)&(x,y)\neq (0,0)\\0&(x,y)=(0,0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec15858e44a62352dc3bf41da37dec4d52a61931)
is Fréchet differentiable and yet fails to have continuous partial derivatives at
Example in infinite dimensions
One of the simplest (nontrivial) examples in infinite dimensions, is the one where the domain is a Hilbert space (
) and the function of interest is the norm. So consider
First assume that
Then we claim that the Fréchet derivative of
at
is the linear functional
defined by
![{\displaystyle Dv:=\left\langle v,{\frac {x}{\|x\|}}\right\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90e32b032fb05acdc2533a8218e5547be8951f68)
Indeed,
![{\displaystyle {\begin{aligned}{\frac {|\|x+h\|-\|x\|-Dh|}{\|h\|}}&={\frac {|\|x\|\|x+h\|-\langle x,x\rangle -\langle x,h\rangle |}{\|x\|\|h\|}}\\[8pt]&={\frac {|\|x\|\|x+h\|-\langle x,x+h\rangle |}{\|x\|\|h\|}}\\[8pt]&={\frac {|\langle x,x\rangle \langle x+h,x+h\rangle -\langle x,x+h\rangle ^{2}|}{\|x\|\|h\|(|\|x\|\|x+h\|+\langle x,x+h\rangle |)}}\\[8pt]&={\frac {\langle x,x\rangle \langle h,h\rangle -\langle x,h\rangle ^{2}}{\|x\|\|h\|(|\|x\|\|x+h\|+\langle x,x+h\rangle |)}}\\&{}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c35a1592e1c1d1b3306ed970cea8d153de659227)
Using continuity of the norm and inner product we obtain:
![{\displaystyle {\begin{aligned}\lim _{\|h\|\to 0}{\frac {|\|x+h\|-\|x\|-Dh|}{\|h\|}}&=\lim _{\|h\|\to 0}{\frac {\langle x,x\rangle \langle h,h\rangle -\langle x,h\rangle ^{2}}{\|x\|\|h\|(|\|x\|\|x+h\|+\langle x,x+h\rangle |)}}\\[8pt]&={\frac {1}{2\|x\|^{3}}}\lim _{\|h\|\to 0}{\frac {\langle x,x\rangle \langle h,h\rangle -\langle x,h\rangle ^{2}}{\|h\|}}\\[8pt]&={\frac {1}{2\|x\|^{3}}}\lim _{\|h\|\to 0}\left(\langle x,x\rangle \|h\|-\langle x,h\rangle \left\langle x,{\frac {h}{\|h\|}}\right\rangle \right)\\[8pt]&={\frac {1}{2\|x\|^{3}}}\left(\lim _{\|h\|\to 0}\langle x,x\rangle \|h\|-\lim _{h\to 0}\langle x,h\rangle \left\langle x,{\frac {h}{\|h\|}}\right\rangle \right)\\[8pt]&={\frac {1}{2\|x\|^{3}}}\left(0-\lim _{\|h\|\to 0}\langle x,h\rangle \left\langle x,{\frac {h}{\|h\|}}\right\rangle \right)\\[8pt]&=-{\frac {1}{2\|x\|^{3}}}\left(\lim _{\|h\|\to 0}\langle x,h\rangle \left\langle x,{\frac {h}{\|h\|}}\right\rangle \right)\\[8pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e330f412e9d74aa04ffdd73fefcf88f0b9a86b1a)
As
and because of the Cauchy-Schwarz inequality
![{\displaystyle \left\langle x,{\frac {h}{\|h\|}}\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e0aee1b6abc66d8b3b68c7c7b2b38b403ddf03)
is bounded by
![{\displaystyle \|x\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be45ae82237598ea927f321e2e1e04e36633b93b)
thus the whole limit vanishes.
Now we show that at
the norm is not differentiable, that is, there does not exist bounded linear functional
such that the limit in question to be
Let
be any linear functional. Riesz Representation Theorem tells us that
could be defined by
for some
Consider
![{\displaystyle A(h)={\frac {|\|0+h\|-\|0\|-Dh|}{\|h\|}}=\left|1-\left\langle a,{\frac {h}{\|h\|}}\right\rangle \right|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e79c4007bf64a1608bccf44a7aa0b848f70d1c5a)
In order for the norm to be differentiable at
we must have
![{\displaystyle \lim _{\|h\|\to 0}A(h)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7cbdd81c5d2b7cb022e23e849ef992701ae58e2)
We will show that this is not true for any
If
obviously
independently of
hence this is not the derivative. Assume
If we take
tending to zero in the direction of
(that is,
where
) then
hence
![{\displaystyle \lim _{\|h\|\to 0}A(h)\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a82c4562d7d45f8e7a797076a3934b58d4e74b9)
(If we take
tending to zero in the direction of
we would even see this limit does not exist since in this case we will obtain
).
The result just obtained agrees with the results in finite dimensions.
Relation to the Gateaux derivative
A function
is called Gateaux differentiable at
if
has a directional derivative along all directions at
This means that there exists a function
such that
![{\displaystyle g(v)=\lim _{t\to 0}{\frac {f(x+tv)-f(x)}{t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7147dda95a84b858872ff080b0495065a4ee2170)
for any chosen vector
![{\displaystyle v\in V,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0415927601750a7c5a284675cdf3b50c6481c294)
and where
![{\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
is from the scalar field associated with
![{\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
(usually,
![{\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
is
real).
[1] If
is Fréchet differentiable at
it is also Gateaux differentiable there, and
is just the linear operator
However, not every Gateaux differentiable function is Fréchet differentiable. This is analogous to the fact that the existence of all directional derivatives at a point does not guarantee total differentiability (or even continuity) at that point. For example, the real-valued function
of two real variables defined by
![{\displaystyle f(x,y)={\begin{cases}{\frac {x^{3}}{x^{2}+y^{2}}}&(x,y)\neq (0,0)\\0&(x,y)=(0,0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b28d758f95d89cba0dd2e1ca92bb5a956e072f7)
is continuous and Gateaux differentiable at the origin
![{\displaystyle (0,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a)
, with its derivative at the origin being
![{\displaystyle g(a,b)={\begin{cases}{\frac {a^{3}}{a^{2}+b^{2}}}&(a,b)\neq (0,0)\\0&(a,b)=(0,0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ceef64f258e2c029100ba248b095e6f0f48d2e0)
The function
is not a linear operator, so this function is not Fréchet differentiable.
More generally, any function of the form
where
and
are the polar coordinates of
is continuous and Gateaux differentiable at
if
is differentiable at
and
but the Gateaux derivative is only linear and the Fréchet derivative only exists if
is sinusoidal.
In another situation, the function
given by
![{\displaystyle f(x,y)={\begin{cases}{\frac {x^{3}y}{x^{6}+y^{2}}}&(x,y)\neq (0,0)\\0&(x,y)=(0,0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85b379af5dfaea460b16d6fec30098afe85c73ac)
is Gateaux differentiable at
![{\displaystyle (0,0),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/463191f337fe0d5d52ed35adfaf91da46fb3a984)
with its derivative there being
![{\displaystyle g(a,b)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6dd3374521de1d53033b0cfafbd40afb41d8a46)
for all
![{\displaystyle (a,b),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0066e8fc90702a659ee69bff970050aba59ee02c)
which
is a linear operator. However,
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is not continuous at
![{\displaystyle (0,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a)
(one can see by approaching the origin along the curve
![{\displaystyle \left(t,t^{3}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2596827ce99f5134ea34040cb7290bf7ae0adda8)
) and therefore
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
cannot be Fréchet differentiable at the origin.
A more subtle example is
![{\displaystyle f(x,y)={\begin{cases}{\frac {x^{2}y}{x^{4}+y^{2}}}{\sqrt {x^{2}+y^{2}}}&(x,y)\neq (0,0)\\0&(x,y)=(0,0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/080ae58279afccfef23971ed86b3fd47fcf8642d)
which is a continuous function that is Gateaux differentiable at
![{\displaystyle (0,0),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/463191f337fe0d5d52ed35adfaf91da46fb3a984)
with its derivative at this point being
![{\displaystyle g(a,b)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6dd3374521de1d53033b0cfafbd40afb41d8a46)
there, which is again linear. However,
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is not Fréchet differentiable. If it were, its Fréchet derivative would coincide with its Gateaux derivative, and hence would be the zero operator
![{\displaystyle A=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75c34024483e6fb7c89e45aff3882ebf11d95a00)
; hence the limit
![{\displaystyle \lim _{\|h\|_{2}\to 0}{\frac {|f((0,0)+h)-f(0,0)-Ah|}{\|h\|_{2}}}=\lim _{h=(x,y)\to (0,0)}\left|{\frac {x^{2}y}{x^{4}+y^{2}}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa1b2143d84a0d53771e5c1da45e152a591488b)
would have to be zero, whereas approaching the origin along the curve
![{\displaystyle \left(t,t^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61f322e3e865508be58c9d3efef3e9dd2023aa2d)
shows that this limit does not exist.
These cases can occur because the definition of the Gateaux derivative only requires that the difference quotients converge along each direction individually, without making requirements about the rates of convergence for different directions. Thus, for a given ε, although for each direction the difference quotient is within ε of its limit in some neighborhood of the given point, these neighborhoods may be different for different directions, and there may be a sequence of directions for which these neighborhoods become arbitrarily small. If a sequence of points is chosen along these directions, the quotient in the definition of the Fréchet derivative, which considers all directions at once, may not converge. Thus, in order for a linear Gateaux derivative to imply the existence of the Fréchet derivative, the difference quotients have to converge uniformly for all directions.
The following example only works in infinite dimensions. Let
be a Banach space, and
a linear functional on
that is discontinuous at
(a discontinuous linear functional). Let
![{\displaystyle f(x)=\|x\|\varphi (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/746e4980bf7b22746dfd4da0a1f31cdc22f6dc25)
Then
is Gateaux differentiable at
with derivative
However,
is not Fréchet differentiable since the limit
![{\displaystyle \lim _{x\to 0}\varphi (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4651e52e4f2bdac10cfc1025b7f69c193cbaa6)
does not exist.
Higher derivatives
If
is a differentiable function at all points in an open subset
of
it follows that its derivative
![{\displaystyle Df:U\to L(V,W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fcf21e0da501a70b78b91699518362a25d051c9)
is a function from
![{\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
to the space
![{\displaystyle L(V,W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e7784b18512cb31d0952dab18cc9612caea99e)
of all bounded linear operators from
![{\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
to
![{\displaystyle W.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/035a080f11445ba1e7745704a6031989a311a7d7)
This function may also have a derivative, the
second order derivative of
![{\displaystyle f,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9687ea22c0f310582e97ee5f6c6a5fca28203d)
which, by the definition of derivative, will be a map
![{\displaystyle D^{2}f:U\to L(V,L(V,W)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b18a813e22fced872241aa632896a55da6f24c)
To make it easier to work with second-order derivatives, the space on the right-hand side is identified with the Banach space
of all continuous bilinear maps from
to
An element
in
is thus identified with
in
such that for all
![{\displaystyle \varphi (x)(y)=\psi (x,y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a119109f23136d4c7dbb4749c95b0a5f4ca84c)
(Intuitively: a function
linear in
with
linear in
is the same as a bilinear function
in
and
).
One may differentiate
![{\displaystyle D^{2}f:U\to L^{2}(V\times V,W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1930f453db5399402dcb9b042394207db6110ba)
again, to obtain the
third order derivative, which at each point will be a
trilinear map, and so on. The
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-th derivative will be a function
![{\displaystyle D^{n}f:U\to L^{n}(V\times V\times \cdots \times V,W),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3e21bf21763346b9c1133674fa03929c95bbdb)
taking values in the Banach space of continuous
multilinear maps in
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
arguments from
![{\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
to
![{\displaystyle W.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/035a080f11445ba1e7745704a6031989a311a7d7)
Recursively, a function
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
is
![{\displaystyle n+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1)
times differentiable on
![{\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
if it is
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
times differentiable on
![{\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
and for each
![{\displaystyle x\in U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906)
there exists a continuous multilinear map
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
of
![{\displaystyle n+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1)
arguments such that the limit
![{\displaystyle \lim _{h_{n+1}\to 0}{\frac {\left\|D^{n}f\left(x+h_{n+1}\right)(h_{1},h_{2},\ldots ,h_{n})-D^{n}f(x)(h_{1},h_{2},\ldots ,h_{n})-A\left(h_{1},h_{2},\ldots ,h_{n},h_{n+1}\right)\right\|}{\left\|h_{n+1}\right\|}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/765f32e6919351e8f3480c6d73e7b84a75c86daa)
exists
uniformly for
![{\displaystyle h_{1},h_{2},\ldots ,h_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceee8aa67e9bfc65bb5e4badb876990ca5ed5f2f)
in bounded sets in
![{\displaystyle V.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978)
In that case,
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
is the
![{\displaystyle (n+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b30a29cfd35628469f9dbffea4804f5b422f3037)
st derivative of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
at
Moreover, we may obviously identify a member of the space
with a linear map
through the identification
thus viewing the derivative as a linear map.
Partial Fréchet derivatives
In this section, we extend the usual notion of partial derivatives which is defined for functions of the form
to functions whose domains and target spaces are arbitrary (real or complex) Banach spaces. To do this, let
and
be Banach spaces (over the same field of scalars), and let
be a given function, and fix a point
We say that
has an i-th partial differential at the point
if the function
defined by
![{\displaystyle \varphi _{i}(x)=f(a_{1},\ldots ,a_{i-1},x,a_{i+1},\ldots a_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54facde8d0ef42f80e8a12f0a7764d700fe133aa)
is Fréchet differentiable at the point
![{\displaystyle a_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f)
(in the sense described above). In this case, we define
![{\displaystyle \partial _{i}f(a):=D\varphi _{i}(a_{i}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a757d1d43fd5a7802f017112de7282f77edd028)
and we call
![{\displaystyle \partial _{i}f(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/834cdef373638311c1bdd9cf609dd5eff2611869)
the i-th partial derivative of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
at the point
![{\displaystyle a.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b803da9c45c1186883bde55107e9ccb102c92c6)
It is important to note that
![{\displaystyle \partial _{i}f(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/834cdef373638311c1bdd9cf609dd5eff2611869)
is a linear transformation from
![{\displaystyle V_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c)
into
![{\displaystyle W.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/035a080f11445ba1e7745704a6031989a311a7d7)
Heuristically, if
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
has an i-th partial differential at
![{\displaystyle a,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f059f053fcf9f421b7c74362cf3bd5ed024e19d1)
then
![{\displaystyle \partial _{i}f(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/834cdef373638311c1bdd9cf609dd5eff2611869)
linearly approximates the change in the function
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
when we fix all of its entries to be
![{\displaystyle a_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0096fb78d6843c9fb67a840dc796b61ad93eec2)
for
![{\displaystyle j\neq i,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f61b97ee36ba71045cff55b8c9b460e962eae9fb)
and we only vary the i-th entry. We can express this in the Landau notation as
![{\displaystyle f(a_{1},\ldots ,a_{i}+h,\ldots a_{n})-f(a_{1},\ldots ,a_{n})=\partial _{i}f(a)(h)+o(h).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c6ca44ca10e3da8f5ab5e4e21339d238c5c924)
Generalization to topological vector spaces
The notion of the Fréchet derivative can be generalized to arbitrary topological vector spaces (TVS)
and
Letting
be an open subset of
that contains the origin and given a function
such that
we first define what it means for this function to have 0 as its derivative. We say that this function
is tangent to 0 if for every open neighborhood of 0,
there exists an open neighborhood of 0,
and a function
such that
![{\displaystyle \lim _{t\to 0}{\frac {o(t)}{t}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af1da34616241aa206f6c4fdb535fe75c5cff499)
and for all
![{\displaystyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
in some neighborhood of the origin,
We can now remove the constraint that
by defining
to be Fréchet differentiable at a point
if there exists a continuous linear operator
such that
considered as a function of
is tangent to 0. (Lang p. 6)
If the Fréchet derivative exists then it is unique. Furthermore, the Gateaux derivative must also exist and be equal the Fréchet derivative in that for all
![{\displaystyle \lim _{\tau \to 0}{\frac {f(x_{0}+\tau v)-f(x_{0})}{\tau }}=f'(x_{0})v,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45f418779d05056707857a5ac70785ee22b28299)
where
![{\displaystyle f'(x_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc15f7bc4034ace9faccf92eb8e3f245541c5e6e)
is the Fréchet derivative. A function that is Fréchet differentiable at a point is necessarily continuous there and sums and scalar multiples of Fréchet differentiable functions are differentiable so that the space of functions that are Fréchet differentiable at a point form a subspace of the functions that are continuous at that point. The chain rule also holds as does the Leibniz rule whenever
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
is an algebra and a TVS in which multiplication is continuous.
See also
Notes
- ^ It is common to include in the definition that the resulting map
must be a continuous linear operator. We avoid adopting this convention here to allow examination of the widest possible class of pathologies.
References
- Cartan, Henri (1967), Calcul différentiel, Paris: Hermann, MR 0223194.
- Dieudonné, Jean (1969), Foundations of modern analysis, Boston, MA: Academic Press, MR 0349288.
- Lang, Serge (1995), Differential and Riemannian Manifolds, Springer, ISBN 0-387-94338-2.
- Munkres, James R. (1991), Analysis on manifolds, Addison-Wesley, ISBN 978-0-201-51035-5, MR 1079066.
- Previato, Emma, ed. (2003), Dictionary of applied math for engineers and scientists, Comprehensive Dictionary of Mathematics, London: CRC Press, ISBN 978-1-58488-053-0, MR 1966695.
- Coleman, Rodney, ed. (2012), Calculus on Normed Vector Spaces, Universitext, Springer, ISBN 978-1-4614-3894-6.
External links
- B. A. Frigyik, S. Srivastava and M. R. Gupta, Introduction to Functional Derivatives, UWEE Tech Report 2008-0001.
- http://www.probability.net. This webpage is mostly about basic probability and measure theory, but there is nice chapter about Frechet derivative in Banach spaces (chapter about Jacobian formula). All the results are given with proof.
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