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Use the compactness and contradiction method to derive finitary quantitative results from infinitary qualitative ones

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Quick description

Suppose one wants to show that all objects A in some class obey some property P_N, if N is a sufficiently large parameter (independent of A). Then one way to establish such a result is by a "compactness and contradiction" argument, which runs as follows:

  • Suppose for contradiction that the claim is false. Thus, no matter how large one takes N, one can find an object A_N which fails to obey the property P_N.

  • Use some sort of compactness theorem to show that some subsequence A_{N_j} of these objects converges (in a suitable topology) to a limit object A_\infty.

  • Show that the limit object A_\infty obeys some limit property P_\infty.

  • Deduce that the approximating objects A_{N_j} obey the property P_{N_j} for sufficiently large j, obtaining the desired contradiction.

The original claim may involve "finitary" objects A (e.g. finite sets of integers, finite graphs, discrete random variables, etc.), but in order to obtain compactness, the limit object A_\infty usually needs to live in some "infinitary" class (e.g. infinite sets of integers, infinite graphs, continuous random variables, etc.). On the other hand, while the original properties P_N to be proved tend to be of a fairly "quantitative" nature (due to the presence of the parameter N), the limit property P_\infty can be of a "qualitative" nature. Thus the compactness method allows one to trade a finitary, quantitative problem for an infinitary, qualitative one, thus bringing to bear the power of "soft analysis" tools such as measure theory, ergodic theory, topology, etc. However, one drawback of the compactness method is that it becomes rather difficult to extract explicit bounds on the parameter N that one obtains at the end of the day, due to the indirect nature of the argument (or more precisely, one can eventually obtain bounds by a large amount of effort, but they tend to be quite poor - tower-exponential bounds, for instance, are quite frequent.)

Prerequisites

Analysis

Example 2

(Discuss minimal-energy blowup solutions in critical PDE)

General discussion

In order to get a sufficient amount of compactness, it is often necessary to settle for convergence in a weak topology, so that results such as the Banach-Alaoglu theorem, Prokhorov's theorem, or the Arzela-Ascoli theorem can be applied. In particular, the use of diagonalization arguments to extract the convergent subsequence are common; see "convergent subsequences and diagonalization" for further discussion. However, in some cases one can later try to "upgrade" the weak convergence to a stronger notion of convergence.

Instead of taking a convergent subsequence A_{N_j} of the original sequence A_N, one can also try taking an ultralimit or ultraproduct of the A_N, or to transfer the problem to a non-standard model in which N becomes a non-standard parameter, larger than all standard parameters.

Comments

There is some overlap between

There is some overlap between this and convergent subsequences and diagonalization. Perhaps some interlinking or merging would be appropriate.

Looks like this article is basically a special case

... of the diagonalisation article, so I've interlinked and parented accordingly.