1. Introduction
The concept of a derivative is one of the most fundamental in mathematics. Its definition is based on the concept of the limit of a function [
1]. However, it is known that a formulation equivalent to the concept of a derivative can be obtained without having to use the concept of a limit. For example, in [
2] an equivalent version of the derivative is presented, using only the topological concept of continuity and the algebraic concept of “factorization”, which in his honor is known as the
Carathéodory derivative.In [
3], Kuhn again takes up the definition given in [
2] and shows the strength of this definition in the proof of the classical theorems of differential calculus. In [
4], the concept of a derivative is presented for functions defined between pseudo-topological vector spaces through the use of Filters Theory, and the basic properties of the derivative are demonstrated.
The Carathéodory derivative has been generalized to functions
[
5], where the equivalence between the Fréchet derivative and Carathéodory derivative is proved as well as the Inverse Function Theorem. The Carathéodory derivative has been defined for functions between Banach spaces [
6,
7]. For example, in [
6], using this generalization, Schwarz’s Lemma on the equality of mixed partial derivatives and Banach’s Fixed Point Theorem are proved, and in [
7], the derivative of Carathéodory allows the determination of the derivative of certain norms in function spaces.
In [
8], the equivalence in
between Gateaux, Fréchet and Carathéodory derivatives is presented using a special topology in
that guarantees the continuity of the Gateaux derivative (radial topology). Since the derivative of Carathéodory involves the topological components of continuity and algebraic of factorization, in [
9] the derivative of Carathéodory is generalized functions with a domain and co-domain as topological groups, showing among other things the important chain rule, and sufficient conditions are given to the uniqueness of the derivative.
Let us remember that a topological group or continuous group results from the union of two important mathematical structures: the topological space and the group. The union is in the sense that the group operation and the inversion operation must be continuous with the given topology [
10,
11,
12].
If in the definition of topological group we consider the metric structure in exchange for the topological structure [
13], we then obtain a
metric group, so it is possible to also have a definition of the Carathéodory differentiability for functions between metric groups. In [
14], the definition of differentiability given in [
9] is used, and a derivative is defined in metric groups. However, in [
14], it was not possible to prove the chain rule.
However, in [
15], a result related to the convergence of certain homomorphisms is presented, which is the key to being able to demonstrate the chain rule, and it is what we will show in this work.
In this paper, we return to the definition of differentiability between metric groups given in [
14], and we endow the
space with an adequate metric, showing the chain rule and the linearity of the derivative. In particular, to show the homogeneity of the derivative of a function between metric groups, we must choose the co-domain of that function as a topological vector space.
2. Group of Continuous Homomorphisms and Group Metric
In this section, we will endow the space of continuous homomorphisms with a metric and a binary operation so that this space is a group. The above will allow us to construct a derivative in metric groups.
The space of continuous homomorphisms between metric groups is defined as:
where
e represents the identity element of the metric group
G.
As the application defined as belongs to , it follows that .
The next definition is a generalization of the one presented in [
13] (p. 159), and we will use it to prove Propositions 1 and 5.
Definition 1. Let be a metric group and the identity element. We define by group metric the metric of H, if for k fixed in H and for allwe have:
- (1)
.
- (2)
There exists such that .
As examples of group metrics, we have, among others, the metric induced by the usual norm in
, and the metric induced by the usual norm in
. In
, we define the usual metric as follows:
Here
is the Euclidean norm. Therefore,
d is a metric group. To see this, one can observe that:
On the other hand, if
K is fixed in
, we have that
Therefore, condition
in Definition 1 is satisfied if we let
.
In what follows, we consider H as an Abelian metric group with a group metric.
Definition 2. In the set , we define the following metric: Next we will endow with a binary operation, and later we will show that this space with this binary operation is an Abelian group.
Definition 3. We define in the following binary operation:where is defined as and ∗ is the binary operation on H. Proposition 1. ⊕ is well defined.
Proof. Let , such that . Therefore, y . For , we have that , i.e., , or . If , then of course . As and are functions, then and , and therefore, .
Next, we claim that . Let be. Then:
As , and are continuous. Let be and . For there are such that:
implies and
implies .
If we do
, then
implies:
The next proposition will be used to prove the existence of invertible elements in
, and its proof is similar to the one presented in [
6].
Proposition 2. Let and be a continuous function in . Then
The proof follows from the continuity of at and continuity of h at a.
Proposition 3. is Abelian group.
In effect, the application defined as for all , is the module of . The function defined by is the inverse of . We also have that is a homomorphism of G in H. The associativity and commutativity in H allow us to see that the operation ⊕ is associative and commutative.
Furthermore, since
G is a metric group then the
defined by
function is continuous, and by the Proposition 2 statement we have:
We have thus shown that .
We use the following theorem to prove the linearity of the derivative.
Theorem 1 ([
15])
. Let G and H be metric groups, G locally compact, . Let be fixed. Thenimpliesthat is, ϕ is continuous at a. 3. Construction of a Derivative in Metric Groups
Definition 4 ([
9])
. Let G and H be metric groups. A function is differentiable at if there is a neighborhood of a and a function continuous at a such that: The derivative of f at a is , and is a slope function for f at a. The latter is Carathéodory’s characterization of differentiability between metric groups.
In this definition, is endowed with the metric given in Definition (2).
To guarantee the uniqueness of the derivative of a function , we use the following definition.
Definition 5 ([
14])
. A group G is divisible if for every element and every there exists such that the equation has only one solution.Furthermore, we impose the condition for any .
If the operation of the group G is given in additive form, we will write and .
For example, are divisible groups.
The next proposition will help us to show the uniqueness of the derivative.
Proposition 4. Let G be a divisible metric group and continues on . Then
Indeed, since the defined by is continuous at G (because it is homeomorphism) and is continuous at a, the result follows as G is divisible.
Regarding the uniqueness of the derivative, we have the following result, proof of which can be found in [
14].
Theorem 2. Uniqueness of the derivativeLet G be a divisible metric group, H Abelian metric group with a group metric. If is differentiable at , then is unique.
Proposition 5 and Theorem 3 will allow us to show that differentiability implies continuity.
Proposition 5. Let be continuous at and . Then
Proof. Since
H has a group metric, for
and
there is
such that
Because φ is continuous at a, for
there is
such that:
i.e.,
□
The proof of Theorem 3 is similar to those in [
6] and will be omitted.
Theorem 3. Let be a continuous function at , continuous at a. Then
We will use Theorem 4 to prove the additivity of the derivative.
Theorem 4. Let be continuous in , , the product metric defined by If and , then
Proof. Since
H is a metric space, we have that:
Let
be fixed. If we perform in the previous limit
,
then by the definition of limit, for
there exists
such that:
which implies that
.
Since
are continuous at
a, then for
there exists
such that:
If we let
, then
implies that:
From which
Similarly, implies
Then,
implies:
Therefore, for
there exists
such that
implies
that is,
□
We use Theorem 5 to demonstrate the important chain rule. This proof is analogous to Theorem 4 and is omitted.
Theorem 5. Let M be an Abelian metric group with a group metric, continuous at , continuous at continuous at and continuous at Therefore, for all , we have that: 4. Basic Theorems of Differentiation
In this section, we will demonstrate the basic theorems of differentiation such as: differentiability implies continuity, the linearity of the derivative and the chain rule. To achieve this, we naturally define, for
and
,
, where * represents the binary operation in
H. Although the following theorems appear in [
6], for convenience they are presented here.
Theorem 6. Differentiability implies continuity
Let be differentiable at . Then f is continuous at a.
Proof. Since
f is differentiable at
a, there exists
continuous at
a such that:
for
x in some neighborhood of
a. Then, in this neighborhood, we have:
If we do
and use Proposition (5), we have that:
Next, by Theorem (3) we have that:
□
Theorem 7. Linearity of the derivative
Assume that are differentiable at . Then is differentiable at a and .
Proof. Since
are differentiable at
a, there are
and
neighborhoods of
a and
continuous at
a such that:
Now, for
we have:
As
is a group,
Let
By Theorem (4) we have that
is continuous in
a and by so much.
□
If we want to obtain a version of the homogeneity of the derivative, we must provide the metric group with a multiplication by scalar compatible with the operation of the group. In this way, the metric group is nothing more than a topological vector space with the topology induced by the metric.
Suppose, then that in the metric group a vector space structure K and a continuous application of in continuous H, where in space the product metric is considered.
Note that the vector space structure automatically requires the group to be Abelian.
If H is a topological vector space over the field K and is a function, we define as for .
Theorem 8. Homogeneity of the derivative: Let be a metric group and topological vector space over field K. If f is differentiable in , then is differentiable at a and also Proof. It only remains to be seen that the application defined as is continuous at a, where is continuous at a.
Indeed, given the continuity of in a, we have when , and hence, for , we have that .
Let us now set in K and , then we have that , from where for all . Now, by Theorem 1 it follows that . □
Theorem 9. Chain rule
Let M be an Abelian metric group, differentiable at , differentiable at . Then is differentiable at a and .
Proof. Since f is differentiable at a and g is differentiable at , there are continuous at a and continuous at such that and , where + is the binary operation of M (usual notation due to M is an Abelian group).
Let
. By Theorems 1 and 5, we have that
is continuous on
a and so
□
Notice the impressive proof of the chain rule.
In order to illustrate the exposed theory, we consider two examples, as follows:
Example 1. Let with a usual matrix addition, which is a divisible and Abelian metric group with a usual metric, defined as above. In the first place, let us consider the functionand let us see that it is differentiable at every point A of G. Indeed, letthen we have that is a homomorphism. Indeed, It only remains to show that is continuous at A. By Theorem 1, it is enough to show thatwe have that Therefore, for and , if we perform we have that . We have shown that f is differentiable at A and , where . For , the result is clear.
In this example, the derivative is equal to the Fréchet derivative.
Example 2. Let G be an Abelian topological group andwe have that: Here, is the identity function, is defined by and is defined by . It is clear that both and are continuous. Therefore, function is continuous. Furthermore, because G is an Abelian group, is a homomorphism. In summary, we obtain that . Finally, if defined by were continuous at a (this depends on the metric chosen in G), then function f is differentiable at a in the sense of Definition 4, with .
5. Conclusions
Carathéodory differentiability for functions has the advantage of being able to be generalized in the context of metric groups. In this sense, a definition of differentiability in metric groups has been given, showing its most important properties, such as linearity and the chain rule. The theory is illustrated by calculating the derivative of a specific function between metric groups.