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One of the most useful concepts for analysis that arise from topology and metric spaces is the concept of compactness; recall that a space ${X}$ is compact if every open cover of ${X}$ has a finite subcover, or equivalently if any collection of closed sets with the finite intersection property (i.e. every finite subcollection of these sets has non-empty intersection) has non-empty intersection. In these notes, we explore how compactness interacts with other key topological concepts: the Hausdorff property, bases and sub-bases, product spaces, and equicontinuity, in particular establishing the useful Tychonoff and Arzelá-Ascoli theorems that give criteria for compactness (or precompactness).

Exercise 1 (Basic properties of compact sets)

• Show that any finite set is compact.
• Show that any finite union of compact subsets of a topological space is still compact.
• Show that any image of a compact space under a continuous map is still compact.

Show that these three statements continue to hold if “compact” is replaced by “sequentially compact”.

To progress further in our study of function spaces, we will need to develop the standard theory of metric spaces, and of the closely related theory of topological spaces (i.e. point-set topology).  I will be assuming that students in my class will already have encountered these concepts in an undergraduate topology or real analysis course, but for sake of completeness I will briefly review the basics of both spaces here.

In the previous lecture, we studied the recurrence properties of compact systems, which are systems in which all measurable functions exhibit almost periodicity – they almost return completely to themselves after repeated shifting. Now, we consider the opposite extreme of mixing systems – those in which all measurable functions (of mean zero) exhibit mixing – they become orthogonal to themselves after repeated shifting. (Actually, there are two different types of mixing, strong mixing and weak mixing, depending on whether the orthogonality occurs individually or on the average; it is the latter concept which is of more importance to the task of establishing the Furstenberg recurrence theorem.)

We shall see that for weakly mixing systems, averages such as $\frac{1}{N} \sum_{n=0}^{N-1} T^n f \ldots T^{(k-1)n} f$ can be computed very explicitly (in fact, this average converges to the constant $(\int_X f\ d\mu)^{k-1}$). More generally, we shall see that weakly mixing components of a system tend to average themselves out and thus become irrelevant when studying many types of ergodic averages. Our main tool here will be the humble Cauchy-Schwarz inequality, and in particular a certain consequence of it, known as the van der Corput lemma.

As one application of this theory, we will be able to establish Roth’s theorem (the k=3 case of Szemerédi’s theorem).

I’m continuing my series of articles for the Princeton Companion to Mathematics with my article on compactness and compactification. This is a fairly recent article for the PCM, which is now at the stage in which most of the specialised articles have been written, and now it is the general articles on topics such as compactness which are being finished up. The topic of this article is self-explanatory; it is a brief and non-technical introduction as to the incredibly useful concept of compactness in topology, analysis, geometry, and other areas mathematics, and the closely related concept of a compactification, which allows one to rigorously take limits of what would otherwise be divergent sequences.

The PCM has an extremely broad scope, covering not just mathematics itself, but the context that mathematics is placed in. To illustrate this, I will mention Michael Harris‘s essay for the Companion, ““Why mathematics?”, you may ask“.

I have just uploaded to the arXiv my paper “A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation”, submitted to Dynamics of PDE. This is a short note on one formulation of the Clay Millennium prize problem, namely that there exists a global smooth solution to the Navier-Stokes equation on the torus $({\Bbb R}/{\Bbb Z})^3$ given any smooth divergence-free data. (I should emphasise right off the bat that I am not claiming any major breakthrough on this problem, which remains extremely challenging in my opinion.)
This problem is formulated in a qualitative way: the conjecture asserts that the velocity field $u$ stays smooth for all time, but does not ask for a quantitative bound on the smoothness of that field in terms of the smoothness of the initial data. Nevertheless, it turns out that the compactness properties of the periodic Navier-Stokes flow allow one to equate the qualitative claim with a more concrete quantitative one. More precisely, the paper shows that the following three statements are equivalent:

1. (Qualitative regularity conjecture) Given any smooth divergence-free data $u_0: ({\Bbb R}/{\Bbb Z})^3 \to {\Bbb R}^3$, there exists a global smooth solution $u: [0,+\infty) \times ({\Bbb R}/{\Bbb Z})^3 \to {\Bbb R}^3$ to the Navier-Stokes equations.
2. (Local-in-time quantitative regularity conjecture)
Given any smooth solution $u: [0,T] \times ({\Bbb R}/{\Bbb Z})^3 \to {\Bbb R}^3$ to the Navier-Stokes equations with $0 < T \leq 1$, one has the a priori bound$\| u(T) \|_{H^1(({\Bbb R}/{\Bbb Z})^3)} \leq F( \| u(0) \|_{H^1(({\Bbb R}/{\Bbb Z})^3)} )$ for some non-decreasing function $F:[0,+\infty) \to [0,+\infty)$.
3. (Global-in-time quantitative regularity conjecture) This is the same conjecture as 2, but with the condition $0 < T \leq 1$ replaced by $0 < T < \infty$.

It is easy to see that Conjecture 3 implies Conjecture 2, which implies Conjecture 1. By using the compactness of the local periodic Navier-Stokes flow in $H^1$, one can show that Conjecture 1 implies Conjecture 2; and by using the energy identity (and in particular the fact that the energy dissipation is bounded) one can deduce Conjecture 3 from Conjecture 2. The argument uses only standard tools and is likely to generalise in a number of ways, which I discuss in the paper. (In particular one should be able to replace the $H^1$ norm here by any other subcritical norm.)