One of the most successful and widely applied technique in analysis is the notion of proof by approximation. To prove that a statement P is true in a structure E it is often shown that the approximations of P hold in E then special properties of the structure (i.e., compactness) are cited to conclude that the statement is true.
A systematic treatment of the above idea from the point of view of mathematical logic was started in the 70's. Henson introduced a positive first order logic LPB and a notion of approximate satisfaction ( |=AP) in metric spaces for formulas in this logic. Henson proved that for every sentence P in LPB and every model E the following weak approximation principle was true:
If E |= P then E |=AP P
Originally invented to characterize finite representability in Banach spaces (See Henson), it was later remarked that |=AP possessed all the nice properties from a model theoretic point of view: Compactness, Lowenheim-Skolem theorems, Keisler-Shelah Ultrapower Theorem, etc. By comparison, the usual (and stronger) notion of satisfaction restricted to metric spaces lacks many of the above properties. An active area of research is the model theory of analytic structures for the notion of |=AP (see Henson& Iovino for an introduction).
At the same time Anderson remarked that for a logic similar to LPB , |= and |=AP agreed for special metric structures having compact predicates. He conjectured that this remark could provide a way for obtaining existence theorems in analysis. Specifically, in order to prove that a property P is true in one of the special metric structures (X,d), it is enough to prove that it is approximately valid ((X,d) |=AP P). Let us call a model E rich for a logic L if for every sentence P in L, E |=AP P implies that E |= P. In a series of papers, Fajardo&Keisler exploited this idea extending |=AP to a logic closed under countable conjunction, finite disjunction and bounded existential and universal quantification. They showed that for many metric structures obtained from Nonstandard Analysis are rich. It follows that these structures are ideal to perform proof by approximation using sentences in the above logic: It is enough to prove that the approximations of the desired sentence hold in the structure to be able to conclude that the sentence is true. The usefulness of this new approach was tested by obtaining new existence results for stochastic differential equations.
It is clear from the above account that the study of approximation principles from the point of view of model theory can yield useful results in analysis. We can summarize the main directions of research as follows:
Of particular interest to me is the last item, because such a relationship is of interest in fields like geometry of Banach spaces (see for example Casazza).In particular, I want to look at the situation where one knows that a property P is approximately true in all the models of a class, and we want to know if this implies that the property is true in all the models of the class. Typical examples of relevant questions that fit this pattern arise in the area of fixed point theory (see Goebel&Kirk ). Another example of a relevant question of the above form concerns the distortion problem in Banach spaces (see for example Milman&Tomczak-Jaegermann). It is known that in every Banach space is approximately true that the answer to the distortion problem is negative. Nevertheless, Odell&Schlumprecht proved that the lp (p >1) spaces are distortable. An open question is whether every uniformly convex space is arbitrary distortable.