I'm trying to understand how to form the ultrapower of a normed space.

I read how to construct the underlying vector space here: http://en.wikipedia.org/wiki/Ultraproduct

But I don't see an obvious norm which can be defined on the resulting equivalence classes.

Is there a standard one which always works?

  • $\begingroup$ Taking the ultraproducts of the norms gives an "ultranorm" (it takes values in the nonstandard reals) and then maybe you can take standard parts? $\endgroup$ – Qiaochu Yuan Jun 22 '13 at 19:29
  • $\begingroup$ First you're going to have to describe how you are treating vector spaces as first-order structures. What is the language you are using? Is it a two-sort language or a one-sort language with just a lot of function symbols? $\endgroup$ – Asaf Karagila Jun 22 '13 at 19:29
  • $\begingroup$ @Qiaochu: Not every non-standard real has a standard part. Consider transfinite ones, what would be their standard part? $\endgroup$ – Asaf Karagila Jun 22 '13 at 19:30
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    $\begingroup$ @Asaf: it seems one should take the ultraproduct in an "internal" fashion; in particular, when taking the product one should restrict as GEdgar does. $\endgroup$ – Qiaochu Yuan Jun 22 '13 at 19:43

Let $I$ be an index set, let $\mathcal U$ be an ultrafilter on $I$. For each $i \in I$ let $(X_i, \|\cdot\|_i)$ be a normed space. Let $X$ be the $l^\infty$-product of the $X_i$, consisting in all families $(x_i)_{i \in I}$ such that $x_i \in X_i$ for all $i$ and $\sup_i \|x_i\|_i < \infty$. The ultraproduct is the quotient $X/N$ where $N$ consists of all $(x_i)$ with $\lim_{i,\mathcal U} \|x_i\|_i = 0$. (That's the limit along an ultrafilter.) In space $X/N$ use norm $\lim_{i,\mathcal U}\|x_i\|_i$.

  • $\begingroup$ Ahh so you are indeed restricting the Cartesian product to elements upon which the norm makes sense. $\endgroup$ – roo Jun 22 '13 at 19:51

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