## Are Banach space ultraproducts stable under infinite sums?

For a pair of Banach spaces $X,Y$ and an ultrafilter $U$ it is easy to find an isomorphism between $(X\oplus Y)_U$ and $X_U\oplus Y_U$. Is this preserved under infinite sums, that is,

Let $X$ be an infinite-dimensional Banach space and let $U$ be an ultrafilter over $\mathbb N$. Do we have

$$\ell_\infty(X_U) \approx [\ell_\infty(X)]_U?$$

$\ell_\infty(X)$ denotes the $\ell_\infty$ sum of countably many copies of $X$.

-
Infinte products, not infinite sums – Yemon Choi Nov 17 at 23:50
What is an infinite product? – Slavoj Žižek Nov 18 at 0:54
Consider a one dimensional $X$. – Bill Johnson Nov 19 at 1:47