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1 Perturbations and small perturbations: definitions and some results
We start our discussion on the Campanato theory of near operators with some preliminary tools.
Let be two sets and let a metric be defined on . If is an injective map, we can define a metric on by putting
Indeed, let be a Cauchy sequence in . By definition of , then is a Cauchy sequence in , and in particular in , which is complete. Thus, there exists which is limit of the sequence . is the limit of in , which completes the proof.
A particular case of the previous statement is when is onto (and thus a bijection) and is complete.
Similarly, if is compact in , then is compact with the metric .
In particular, if is injective then is a perturbation of if is uniformly continuous with respect to the metric induced on by .
In the same hypothesis as in the previous definition, we say that is a small perturbation of if it is a perturbation of constant .
is a small perturbation of .
Then, there exists a unique such that
we note that, by (2), we have
where is the constant of the small perturbation; note that, by the definition of and applying to the first side, the last equation can be rewritten as
in other words, since , is a contraction in the complete metric space ; therefore (by the classical Banach-Caccioppoli fixed point theorem) has a unique fixed point: there exist such that ; by definition of this is equivalent to , and the proof is hence complete. ∎
The hypothesis of the theorem can be generalized as such: let be a set and a metric space (not necessarily complete); let be two mappings from to such that is injective, is complete and ; then there exists such that .
(Apply the theorem using instead of as target space.)
The Banach-Caccioppoli fixed point theorem is obtained when and is the identity.
We can use theorem 1.4 to prove a result that applies to perturbations which are not necessarily small (i.e. for which the constant can be greater than one). To prove it, we must assume some supplemental structure on the metric of : in particular, we have to assume that the metric is invariant by dilations, that is that for each . The most common case of such a metric is when the metric is deduced from a norm (i.e. when is a normed space, and in particular a Banach space). The result follows immediately:
Let be a set and be a complete metric space with a metric invariant by dilations. Let be two mappings from to such that is bijective and is a perturbation of , with constant .
Then, for each there exists a unique such that
We also have the following
Let be a set and be a complete, compact metric space with a metric invariant by dilations. Let be two mappings from to such that is bijective and is a perturbation of , with constant .
Then there exists at least one such that
2 Near operators
A historical remark: Campanato initially introduced the concept in Hilbert spaces; subsequently, it was remarked that most of the theory could more generally be applied to Banach spaces; indeed, it was also proven that the basic definition can be generalized to make part of the theory available in the more general environment of metric vector spaces.
We will here discuss the theory in the case of Banach spaces, with only a couple of exceptions: to see some of the extra properties that are available in Hilbert spaces and to discuss a generalization of the Lax-Milgram theorem to metric vector spaces.
2.1 Basic definitions and properties
Let be a set and a Banach space. Let be two operators from to . We say that is near if and only if there exist two constants and such that, for each one has
In other words, is near if is a small perturbation of for an appropriate value of .
Observe that in general the property is not symmetric: if is near , it is not necessarily true that is near ; as we will briefly see, this can only be proven if , or in the case that is a Hilbert space, by using an equivalent condition that will be discussed later on. Yet it is possible to define a topology with some interesting properties on the space of operators, by using the concept of nearness to form a base.
The core point of the nearness between operators is that it allows us to “transfer” many important properties from to ; in other words, if satisfies certain properties, and is near , then satisfies the same properties. To prove this, and to enumerate some of these “nearness-invariant” properties, we will emerge a few important facts.
In what follows, unless differently specified, we will always assume that is a set, is a Banach space and are two operators from to .
If is near then there exist two positive constants such that
which is the first inequality with (which is positive since ).
which is the second inequality with . ∎
The most important corollary of the previous lemma is the following
If is near then two points of have the same image under if and only if the have the same image under .
We can express the previous concept in the following formal way: for each in there exist in such that and conversely. In yet other words: each fiber of is a fiber (for a different point) of , and conversely.
It is therefore possible to define a map by putting ; the range of is . Conversely, it is possible to define , by putting ; the range of is . Both maps are injective and, if restricted to their respective ranges, one is the inverse of the other.
Also observe that and are continuous. This follows from the fact that for each one has
and that the lemma ensures that given a sequence in , the sequence converges to if and only if converges to .
We can now list some invariant properties of operators with respect to nearness. The properties are given in the form “if and only if” because each operator is near itself (therefore ensuring the “only if” part).
a map is injective if and only if it is near an injective operator;
a map is surjective if and only if it is near a surjective operator;
a map is open if and only if it is near an open map;
a map has dense range if and only if it is near a map with dense range.
Another important property that follows from the lemma is that if there exist such that , then it is : intersecting fibers are equal. (Campanato only stated this property for the case and called it “‘the kernel property”; I prefer to call it the “fiber persistence” property.)
2.1.1 A topology based on nearness
Let be the set of maps between and . For each and for each we let the set of all maps such that is a small perturbation of with constant . In other words, if and only if is near with constants .
The set satisfies the axioms of the set of fundamental neighbourhoods. Indeed:
belongs to each ;
for each there exist such that for each there exist .
This last property (permanence of neighbourhoods) is somewhat less trivial, so we shall now prove it.
Let be given.
Let be another arbitrary neighbourhood of and let be an arbitrary element in it. We then have:
but also (lemma 2.2)
Let also be an arbitrary neighbourhood of and an arbitrary element in it. We then have:
The nearness between and is calculated as such:
We then want , that is ; the condition is always satisfied on the right side, and the left side gives us . ∎
It is important to observe that the topology generated this way is not a Hausdorff topology: indeed, it is not possible to separate and (where and is a constant element of ). On the other hand, the subset of all maps with with a fixed valued at a fixed point () is a Hausdorff subspace.
Another important characteristic of the topology is that the set of invertible operators from to is open in (because a map is invertible if and only if it is near an invertible map). This is not true in the topology of uniform convergence, as is easily seen by choosing and the sequence with generic element : the sequence converges (in the uniform convergence topology) to , which is invertible, but none of the is invertible. Hence is an element of which is not inside , and is not open.
2.2 Some applications
As we mentioned in the introduction, the Campanato theory of near operators allows us to generalize some important theorems; we will now present some generalizations of the Lax-Milgram theorem, and a generalization of the Riesz representation theorem.
|Date of creation||2013-03-22 14:03:21|
|Last modified on||2013-03-22 14:03:21|
|Last modified by||mathcam (2727)|
|Synonym||Campanato theory of near operators|