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?



Ultraproducts of Banach spaces versus model theoretic ultraproduct
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](https://faculty.math.illinois.edu/%7Ehenson/cfo/mtfms.pdf) 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.


