Once beyond the realm of normed vector spaces, the various ways of defining differentiation diverge. This is particularly evident if one considers the slightly stronger notion of continuous differentiability wherein the assignment of the derivative must also be continuous.
One can make a reasonable start by saying that for a function to be continuously differentiable then it must at least satisfy the notion of Gâteaux differentiability, and one can throw in the requirement that the assignment of the directional derivative be continuous and linear (this is known as Gâteaux–Lévy differentiability). Thus one obtains a map . However, outside the realm of normed vector spaces there is not a unique topology on and thus one can come up with a variety of meanings for the phrase “ is continuous”.
Finite Dimensions
Let us remind ourselves of the situation in finite dimensions.
Definition
A function , where is open, is said to be continuously differentiable, or of class , if there is a continuous map with the property that for each and then
Note that for and there is an open interval with the property that for then and so the limit makes sense.
Infinite Dimensions
In infinite dimensions the difficulty with extending the standard definition is that of the topology on continuous linear maps. This becomes more evident with higher derivatives. Thus the definition depends on such a choice. In addition, one needn’t use a topology but can make sense of the definition with a convergence structure on the space of linear maps.
Definition
Let and be locally convex topological vector spaces. Let be an open set. Let be the space of continuous linear maps from to . Let be a convergence structure on . A continuous function is said to be differentiable of class if there exists a continuous mapping , called the derivative of , such that for every then
We define to be the set of functions which are of class .
Convergence Structures
There are a variety of convergence structures and topologies on that can be used. Some of them with particular properties are gathered in the list below. For these, the notation is condensed slightly as indicated.
In the following, given semi-norms on and on we define a semi-norm on by
: the translation-invariant convergence structure with filters:
if there is a semi-norm on such that for each semi-norm on and there is some with .
: (Marinescu’s convergence structure) the colimit in the category of convergence vector spaces of the following family of spaces. Let denote the family of mappings from the set of continuous semi-norms on to that on . For define:
: the compatible convergence structure with filters:
if for each semi-norm on there is a semi-norm on such that for each there is some with .
: the quasi-bounded convergence structure. This has filters:
if for every quasi-bounded filter on .
Recall that a filter on is quasi-bounded if where is the neighbourhood filter of in .
: the continuous convergence structure. This is the coarsest convergence structure on which makes the evaluation map continuous.
: the topology of uniform convergence on a family of bounded subsets of , in particular:
: the topology of uniform convergence on bounded subsets of ,
: the topology of uniform convergence on compact subsets of ,
: the topology of uniform convergence on finite subsets of .
The relationships between the various definitions of are displayed in the following diagram.
Let us extract some particular cases:
normable, arbitrary:
, , , , are equivalent
, , are equivalent
Banach space, arbitrary:
, , , , are equivalent
, , , are equivalent
Fréchet space, arbitrary:
, are equivalent
, , , are equivalent
Fréchet–Schwartz, arbitrary: , , , , , , are equivalent
finite dimensional and arbitrary, or Fréchet–Schwartz and normable: All equivalent
Chain Rule
An important question to ask of the various definitions of continuously differentiable is whether they satisfy the chain rule. The following result provides the basis for this.
Lemma
Let , , be LCTVS, let and be open sets, let and be convergence structures on and respectively.
Assume that the composition map:
is continuous.
Let and be functions with . Suppose that is of class and of class . Then is of class .
Corollary
The chain rule holds for each of , , , , .
The following partial chain rules also hold:
If is metrisable,
If is metrisable,
Higher Derivatives
A minor wriggle enters the story with higher derivatives due to the fact that the higher derivatives are multilinear maps and so not only are there different notions of convergence to put on these spaces, there are also different possible meanings of the statement that these are continuous. When dealing with one of the topologies (defined by some family of bounded sets), we will end up with derivatives in rather than in the notation of continuous multilinear operator.
However, we can start with a very weak notion to get the ball rolling. Let be the space of all (not necessarily continuous) -linear maps . We equip it with the topology of simple convergence.
Definition
Let and be LCTVS, an open subset, . A function is said to be weakly -times differentiable if there exist functions for such that and for each , , and then
Note that we don’t assume that the are continuous. If some are continuous then some nice properties ensue. For example, if is continuous then each for is totally symmetric.
From the definition of weakly -times differentiable we can define the various classes of continuously -times differentiable. For the definition of see the page on continuous multilinear operator.
Definition
Let and be LCTVS, an open subset, . Let be a family of bounded sets in which covers . A function is said to be differentiable of class if is weakly -times differentiable and such that for then:
, and
is continuous.
Using convergence structures, we have:
Definition
Let and be LCTVS, an open subset, . Let be one of the convergence structures , , , , or on . A function is said to be differentiable of class if is weakly -times differentiable and such that for then:
, and
is continuous.
For fixed , the relationships between the various are the same as for . For varying we have:
Lemma
Let and be LCTVS, an open set. If is of class then is of class . Whence we have:
If is metrisable (resp. Fréchet) and is of class (resp. ) then is of class .