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One of the key difficulties in performing analysis in infinite-dimensional function spaces, as opposed to finite-dimensional vector spaces, is that the Bolzano-Weierstrass theorem no longer holds: a bounded sequence in an infinite-dimensional function space need not have any convergent subsequences (when viewed using the strong topology). To put it another way, the closed unit ball in an infinite-dimensional function space usually fails to be (sequentially) compact.

As compactness is such a useful property to have in analysis, various tools have been developed over the years to try to salvage some sort of substitute for the compactness property in infinite-dimensional spaces. One of these tools is concentration compactness, which was discussed previously on this blog. This can be viewed as a compromise between weak compactness (which is true in very general circumstances, but is often too weak for applications) and strong compactness (which would be very useful in applications, but is usually false), in which one obtains convergence in an intermediate sense that involves a group of symmetries acting on the function space in question.

Concentration compactness is usually stated and proved in the language of standard analysis: epsilons and deltas, limits and supremas, and so forth. In this post, I wanted to note that one could also state and prove the basic foundations of concentration compactness in the framework of nonstandard analysis, in which one now deals with infinitesimals and ultralimits instead of epsilons and ordinary limits. This is a fairly mild change of viewpoint, but I found it to be informative to view this subject from a slightly different perspective. The nonstandard proofs require a fair amount of general machinery to set up, but conversely, once all the machinery is up and running, the proofs become slightly shorter, and can exploit tools from (standard) infinitary analysis, such as orthogonal projections in Hilbert spaces, or the continuous-pure point decomposition of measures. Because of the substantial amount of setup required, nonstandard proofs tend to have significantly more net complexity than their standard counterparts when it comes to basic results (such as those presented in this post), but the gap between the two narrows when the results become more difficult, and for particularly intricate and deep results it can happen that nonstandard proofs end up being simpler overall than their standard analogues, particularly if the nonstandard proof is able to tap the power of some existing mature body of infinitary mathematics (e.g. ergodic theory, measure theory, Hilbert space theory, or topological group theory) which is difficult to directly access in the standard formulation of the argument.

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