Real Analysis – Applications of Model Theory to Analysis

Question

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from algebraic structures (theories of abstract groups, rings, fields) and real and complex analysis (theories of expansions of real and complex fields, and sometimes both).

While relationships with algebra seem quite apparent, I wonder what are some interesting results in real and complex analysis that have nice model-theoretical proofs (or better yet, only model-theoretical proofs are known!)?

Of course, there's nonstandard analysis, but I hope to see some different examples. That said, I wouldn't mind seeing a particularly interesting application of nonstandard analysis. 🙂

I hope the question is at least a little interesting. I have only the very basic knowledge of model theory of that type (and the same applies to nonstandard analysis), so it may seem a little naive, but I got curious, hence the question.

Answer 1

There is a result in functional analysis whose first known proof uses non-standard techniques:

Theorem If a bounded linear operator $ T $ on a Hilbert space $ \mathcal{H} $ is polynomially compact, i.e., $ P(T) $ is compact for some non-zero polynomial $ P $, then $ T $ has an invariant subspace. This means that there is a non-trivial proper subspace $ W $ of $ \mathcal{H} $ such that $ p(T)[W] \subseteq W $.

The proof was given by Allen Bernstein and Abraham Robinson. Their result is significant because it is related to the so-called Invariant-Subspace Conjecture, an important unsolved problem in functional analysis. Paul Halmos, a staunch critic of non-standard analysis, supplied a standard proof of the result almost immediately after reading the pre-print of the Bernstein-Robinson paper. In fact, both proofs were published in the same issue of the Pacific Journal of Mathematics!