Ultraproducts of Banach spaces versus model theoretic ultraproduct

Question

Reading about ultraproducts in model theory and in Banach spaces leads to two distinct definitions. E.g., for an ultrapower given by an ultrafilter $\mu$ on $\mathbb{N}$, both notions of ultrapower are quotients of the space of sequences $\sigma: \mathbb{N} \to \mathbb{R}$.

In model theory. Two sequences $\sigma,\tau$ are equivalent in the case that $\{ i \in \mathbb{N} : \sigma_i = \tau_i \} \in \mu$. For example, the hyperreals are a model-theoretic ultrapower $\lim_\mu \mathbb{R}$.

For Banach spaces (noting we also restrict to bounded sequences). Two sequences $\sigma,\tau$ are equivalent in the case that $\lim_\mu \| \sigma_i - \tau_i \| = 0$. For example, any ultrapower $(\mathbb{R})_\mu$ is isomorphic to $\mathbb{R}$.

Putting aside the difference that in Banach spaces we only look at bounded sequences, these two equivalence relations are not the same. Therefore, it seems confusing to call both constructions an ultraproduct. This leads to my question:

Is there any argument that shows that the difference between both of these constructions is negligible? Perhaps there are a few possible answers:

• There is almost a quotient map $\lim_\mu \mathbb{R} \to (\mathbb{R})_\mu$ for which one could lift any statement about the Banach space ultrapower to the model-theoretic one.
• The additional “non-standard” elements in the model-theoretic object are not useful for Banach spaces. Though in that case, why call the Banach space construction an ultraproduct?!
• There isn’t any formal result relating the two constructions, but they are at least similar-looking.

Are there any perspectives that fit into the first two points — or at least does not fit into the last point?

Answer 1

The ultraproduct of Banach spaces is the ultraproduct in the sense of metric structures in continuous logic. For a nice survey on this topic, see Model theory for metric structures by Ben Yaacov, Berenstein, Henson, and Usvyatsov. The ultraproduct of metric structures is defined in Section 5, pp. 23-27.

The restriction to bounded sequences in the Banach space ultraproduct is connected to the fact that in the definition of a metric structure, each sort carries the structure of a bounded metric space. One way to view a Banach space as a metric structure is to have a sort $B_n$ for each unit ball of radius $n$, together with the inclusion maps $B_n\to B_m$ when $n<m$. Then in the ultraproduct construction, we only ever consider sequences from a fixed sort.

Every structure in the sense of ordinary model theory is a metric structure with the discrete metric, so indeed these two notions of ultraproduct are instances of the same construction.

Answer 2

As a logician, I take the model-theoretic notion of ultraproduct as the primary one, so the following formal connection describes how to get the Banach-space ultraproduct from the model-theoretic one.

Begin with any normed linear space $V$ in the nonstandard universe, for example an ultraproduct of standard Banach spaces. (Here both "linear" and "normed" refer to scalars that are nonstandard real numbers.) The vectors whose norm is finite (i.e., majorized by a standard natural number) form a vector subspace $A$ over the standard real field, but the norms of its vectors are still in the nonstandard real field. The vectors whose norm is infinitesimal (i.e., smaller than every positive standard rational number) form a (real) vector subspace $B$ of $A$. The quotient $Q=A/B$ is another (real) vector space. The nonstandard-real-valued norm is not well-defined on $Q$; vectors in $A$ that differ by an element of $B$ can have different norms in $A$, but they differ only infinitesimally. So we can define a real-valued norm on $Q$ as the standard part of the nonstandard norm of (any representative) in $A$. In this way, $Q$ becomes a real normed vector space.

If $V$ was obtained as the model-theoretic ultraproduct of some Banach spaces, then $Q$ is the Banach space ultraproduct of those spaces. But the construction of $Q$ from $V$ makes sense more generally, starting with any normed vector space in any nonstandard universe.

In fact, it's more general yet. For example, $V$ could be a nonstandard vector space over only the nonstandard rationals, allowing the norm to have nonstandard real values. Then $Q$ will still be a normed vector space over the reals. As an extremely special case, we could take $V$ to be just the nonstandard rationals; the "finite mod infinitesimal" construction above would then produce the standard reals.

Incidentally, notice that the choice of $A$ and $B$ in the construction of $Q$ is just what is needed in order to define a real-valued norm on the quotient $Q=A/B$. The choice of $A$ makes the norms finite; the choice of $B$ makes the ambiguity in the norms of cosets infinitesimal; and so the standard-part gives a well-defined, real-valued norm.

Answer 3

The "Banach space ultraproduct" is also referred to as the nonstandard hull, precisely in order to distinguish it from the model theoretic ultraproduct (which I shall simply call "ultraproduct" here).

The two spaces are loosely related by the short exact sequences $\require{AMScd}$ \begin{CD} @. @. 0\\ @. @. @VVV\\ 0 @>>> o({\mathbb R}) @>>> O({\mathbb R}) @>>{\mathrm{st}}> ({\mathbb R})^\mu @>>> 0 \\ @. @. @VVV \\ @. @. \lim_\mu {\mathbb R} \\ @. @. @VVV \\ @. @. (\lim_\mu {\mathbb R}) / O({\mathbb R}) \\ @. @. @VVV \\ @. @. 0 \end{CD} where $O({\mathbb R})$ is the space of bounded (aka limited) hyperreals, $o({\mathbb R})$ is the subspace (in fact ideal) of infinitesimal hyperreals, $\mathrm{st}$ is the standard part map, and $\lim_\mu {\mathbb R} / O({\mathbb R})$ is the quotient of the hyperreals by the bounded hyperreals (I don't know if there is a consensus name for this space; it describes a special class of cuts of the hyperreals, but does not contain all possible cuts). But as one sees from this diagram, there is no direct relation between the nonstandard hull ${\mathbb R}^\mu$ of ${\mathbb R}$ (which in this locally compact case is in fact isomorphic to ${\mathbb R}$) and the ultraproduct $\lim_\mu {\mathbb R}$. Nevertheless they can be both be viewed as "completions" of $\mathbb{R}$, and are at least analogous even if not related in any category-theoretic sense. (There may however be model-theoretic ways to relate the two.)