Calculus 1 – Derivatives
Calculus 1 – Derivatives

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 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

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

is continuous ( denotes the space of all bounded linear operators from to ). Note that this is not the same as requiring that the map be continuous for each value of (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

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:

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:

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

In order for the norm to be differentiable at we must have

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

(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.

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

for any chosen vector and where is from the scalar field associated with (usually, 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

is continuous and Gateaux differentiable at the origin , with its derivative at the origin being

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

is Gateaux differentiable at with its derivative there being for all which is a linear operator. However, is not continuous at (one can see by approaching the origin along the curve ) and therefore cannot be Fréchet differentiable at the origin.

A more subtle example is

which is a continuous function that is Gateaux differentiable at with its derivative at this point being there, which is again linear. However, is not Fréchet differentiable. If it were, its Fréchet derivative would coincide with its Gateaux derivative, and hence would be the zero operator ; hence the limit

would have to be zero, whereas approaching the origin along the curve 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.

If is a differentiable function at all points in an open subset of it follows that its derivative

is a function from to the space of all bounded linear operators from to This function may also have a derivative, the second order derivative of which, by the definition of derivative, will be a map

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

(Intuitively: a function linear in with linear in is the same as a bilinear function in and ).

One may differentiate

again, to obtain the third order derivative, which at each point will be a trilinear map, and so on. The -th derivative will be a function

taking values in the Banach space of continuous multilinear maps in arguments from to Recursively, a function is times differentiable on if it is times differentiable on and for each there exists a continuous multilinear map of arguments such that the limit

exists uniformly for in bounded sets in In that case, is the st derivative of 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.

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

is Fréchet differentiable at the point (in the sense described above). In this case, we define and we call the i-th partial derivative of at the point It is important to note that is a linear transformation from into Heuristically, if has an i-th partial differential at then linearly approximates the change in the function when we fix all of its entries to be for and we only vary the i-th entry. We can express this in the Landau notation as

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

and for all 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

where 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 is an algebra and a TVS in which multiplication is continuous.

^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. 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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