In the last two posts on spectral methods in dynamics, we’ve used (both explicitly and implicitly) a number of results and a good deal of intuition on function spaces. It seems worth discussing these a little more at length, as a supplement to the weekly seminar posting.
1. Function spaces and extra structure
It is useful to treat real-valued functions (or complex-valued functions, or vector space-valued functions) as elements of a vector space, so that the tools from linear algebra can be applied. Given a set one may consider the vector space of all real-valued functions with domain . If is finite, say with elements, then this is just the familiar vector space . The more interesting examples are when is infinite, and so is infinite-dimensional. We will focus on the case , which is reasonably representative.
Generally speaking, the functions that arise from some application are not entirely arbitrary, but have some degree of regularity — maybe they are continuous, or piecewise continuous, or measurable, or integrable, etc. It turns out that the vector space is “too large” for many applications, and that it is more suitable to consider a smaller space, whose elements are functions with some extra properties. We will consider some of the ways to do this, paying particular attention to how those choices let us recover certain properties of that involve extra structure beyond that of the vector space itself:
- Topology: We know what it means for a sequence to converge to some , and we want a similar notion of convergence in a vector space .
- Metric and norm: We want the notion of convergence to come from a metric (distance function) that is compatible with the vector space structure of — that is, a norm, with respect to which the vector space becomes a Banach space.
- Compactness: A subset of is compact if every sequence in that subset has a convergent subsequence, and this property is important in many applications and proofs. By the Heine–Borel theorem compactness in is equivalent to being closed and bounded. How can we determine when a set of functions in is compact?
2. Continuous functions and Arzelà–Ascoli
The extra structure we seek to place on should leverage some of the extra structure that has, beyond simply being an uncountable set. In particular, we may use either the topology of or Lebesgue measure on to define properties of functions . First we discuss the topological option — later we see what happens when we use the measure-theoretic structure to define the spaces (and others).
The natural space to use is , the space of continuous real-valued functions on , with the norm . The space of continuous functions is complete with respect to this norm, and so we have a Banach space. What about compactness? How do we tell if a set is compact? Of course should be closed, but what else do we need? Boundedness is no longer enough: the unit ball in is not compact, as can be seen by considering the sequence of functions shown in Figure 1.
Fig 1 Uniformly bounded but no convergent subsequence.
- is uniformly bounded: .
- is equicontinuous: for every there exists such that for every and .
Remark 1 The proof that these conditions guarantee compactness uses the following strategy, which it is a useful exercise to complete:
- Given any sequence , use uniform boundedness and a diagonalisation argument to find a subsequence that converges at every rational number. (Or on some other countable dense set.)
- Use equicontinuity to guarantee that is Cauchy for every , and hence converges.
In particular, one can consider the subspace of Hölder continuous functions with exponent — this is a Banach space with norm
When this is the space of Lipschitz functions. If is uniformly bounded in the norm, then it is uniformly bounded in the norm and equicontinuous, and hence it is pre-compact in the norm.
It is important to note here the structure of the last statement — we have two norms, and , such that uniform boundedness in one norm implies pre-compactness in the other. This is the closest that we can come to an infinite dimensional analogue of Heine–Borel: as a consequence of Riesz’s lemma, every infinite-dimensional Banach space has a uniformly bounded sequence with no convergent subsequence.
In our study of spectral methods in dynamics, an important step is always to find two norms with this relationship: uniform boundedness in one implies pre-compactness in the other. We remark that the Arzelà–Ascoli theorem actually gives just a little bit more than this: given a sequence that is uniformly bounded in the norm, pre-compactness only guarantees the existence of a limit point , but in fact the limit point is in as well, because any modulus of continuity for the sequence is also a modulus of continuity for any limit point.
Another important family of function spaces, which leverages not only the topological but also the differentiable structure of the unit interval, are the spaces , defined inductively as
Here need not be an integer (the base case for the induction is ), so for example, for , is the space of differentiable functions whose derivatives are Hölder continuous with exponent . The space becomes a Banach space when endowed with the norm inductively given by
The relationship discussed above between uniform boundedness in one norm and pre-compactness in another can be stated quite generally for this family of norms: uniform boundedness in the norm implies pre-compactness in the norm for any . This relationship is often expressed by saying that “ is compactly embedded in for ”.
3. Lp spaces
In terms of the measure-theoretic structure of the unit interval, the most important function spaces are the spaces
where , and
where is the essential supremum of .
In fact, this definition cheats a little bit, because elements of an space are actually equivalence classes of functions, where two functions are equivalent if they agree on a set of full Lebesgue measure. This throws a small technical monkey wrench into many arguments involving spaces, since strictly speaking an expression like for has no meaning unless it is inside an integral sign. One way to avoid these technicalities is to emphasise the role of elements of not necessary as functions, but rather as linear functionals.
Recall that if is a Banach space, then is the dual space of continuous linear functionals . The spaces have the property that
Thus instead of thinking of a function , we may think of the associated functional in (2), which is obtained by integrating the function against test functions from a suitable space. In this case the space of test functions is taken to be , but there are many other examples we could consider — eventually this leads to the idea of considering distributions in place of functions, but we will not go this far here.
Remark 2 Before moving on, we note that , but is a larger space than .
4. Weak derivatives
An important use of this alternate viewpoint — functions as continuous linear functionals — is to define the weak derivative of a function. If is differentiable, then for any differentiable with , integration by parts gives
Equation (3) characterises the derivative , which motivates the following definition: is the weak derivative of if
where the space of test functions is . Write in this case.
Example 1 The absolute value function has as its derivative the step function . Note that the value of is not uniquely defined because is considered as an element of .
Writing for the step function just described, we see that does not have a weak derivative in . Indeed, this is true for any function with a jump discontinuity.
Using mollifiers one can show that any function can be approximated by (infinitely) differentiable functions such that approximates in . This can be used to show that the usual product rule for derivatives holds for weak derivatives as well: , as long as and both have weak derivatives. The space of functions with a weak derivative in is denoted and is an important example of a Sobolev space. Here the norm is
which can be viewed as an analogue of the definition of the norm in (1). Moreover, just as the unit ball is compact, so also the unit ball is compact, as we will see.
5. Kolmogorov–Riesz compactness theorem
In understanding compactness for subsets of function spaces, it is useful to recall that the Heine–Borel theorem can be generalised to arbitrary complete metric spaces as follows: a set is compact if and only if it is closed and totally bounded. In particular, for Banach spaces, pre-compactness is equivalent to being totally bounded.
The Arzelà–Ascoli theorem gives a necessary and sufficient condition for a set in to be totally bounded (and hence pre-compact). A similar result in the spaces is the Kolmogorov–Riesz compactness theorem — an expository account of this theorem and its relationship to the Arzelà–Ascoli theorem is given in a recent paper by H. Hanche–Olsen and H. Holden, The Kolmogorov–Riesz compactness theorem (Expo. Math. 28 (2010), 385–394).
In our setting (where we consider spaces with respect to a finite measure), the Kolmogorov–Riesz theorem can be stated as follows: a set is totally bounded (in the norm) if and only if
- is bounded, and
- for every there is such that for every and , where .
In other words, to go from bounded to totally bounded one needs the added condition that small changes to the argument result in (uniformly) small changes in the function, with respect to the norm.
Roughly speaking the idea is that if a set can be “approximately embedded” into a totally bounded set, then it must itself be totally bounded — this is Lemma 1 in the paper referred to above. Then the condition on for allows the set to be “approximately embedded” into a bounded set in by averaging over small neighbourhoods in its domain. This is of course a very rough description and one should read the paper for the complete proof and precise formulation of what it means to be “approximately embedded”.
6. Bounded variation and Helly’s theorem
One can use the Kolmogorov–Riesz theorem to show that is compactly embedded in . (This is a special case of the Rellich–Kondrachov theorem.) However, since functions with jump discontinuities are not in , we want to use a bigger function space in order to study spectral properties of the transfer operator.
The definition of weak derivative can be generalised if one is willing to allow to live somewhere besides . Recall that we want to satisfy
for every test function , the space of functions on the interval that vanish at the endpoints. The left-hand side defines a linear functional , and given any we may define as such a linear functional by setting
If , this functional is not given by integration against an function, but now the definition makes sense for any . Moreover, the space of linear functionals on carries a natural norm: the norm of is
A functional is continuous if and only if . Recalling our discussion of bounded variation functions in an earlier post, we see that , and so
The BV norm can be written as . Note that BV is exactly the set of functions for which is a continuous linear functional on .
Helly’s selection theorem states that is compactly embedded in . (This is not to be confused with Helly’s theorem in geometry.) This is a consequence of the Kolmogorov–Riesz compactness theorem, because a relatively straightforward computation shows that
(See Lemma 11 and Theorem 12 in the paper of Hanche–Olsen and Holden referenced above.) We remark that one can also give a direct proof following the hint given in Footnote 8 of Keller and Liverani’s A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps: given , let be the step function that is constant on each dyadic interval , with value equal to the average of on that interval. Then the functions approach in , and the problem reduces to finding a suitable subsequence of step functions.