Suppose we have a Banach space ultraproduct $(E_i)_U$. I can think of three natural objects which are "dual-like":
- $(E_i^*)_U$, the ultraproduct of the duals of the ground spaces.
The space made up of objects $(f_i)_U$ such that:
a) Each $f_i:E_i\rightarrow\mathbb{R}$
b) There is a $B$ such that for every $i$ and every $x$, $|f_i(x)|\leq B||x||$
c) For any $(x_i)_U, (y_i)_U\in (E_i)_U$, $\lim_U||f_i(x_i+y_i)-f_i(x_i)-f_i(y_i)||=0$
d) For any $(x_i)_U\in (E_i)_U$ and any real $\alpha$, $\lim_U ||f_i(\alpha x_i)-\alpha f_i(x_i)||=0$
The ordinary dual $(E_i)_U^*$.
It's easy to see that the space in (1) is contained in the space in (2) which is contained in the space in (3). Furthermore, I think it's well known that (1) is, in general, strictly smaller than (3).
My question is what is known about (2). Does it have a name? I expect that it can be strictly smaller than (3), but are any examples known?
Edit: A little bit of motivation for why I'm thinking about (2). It has some nice properties that (3) doesn't, since its elements can be described: given $f$ in (2), for each $i$ we have a function $f_i$ on $E_i$, and by choosing $i$ large enough, we can ensure that $f_i$ "resembles" an element of the dual of $E_i$.
But (2) should still act a great deal like the true dual. (Formally, suppose $M$ is some model---say, of $ZFC$---where for every Banach space $X$, $\phi(X,X^*)$ holds, and which contains every element of our sequence $(E_i)$. Then in $(M)_U$, $(E_i)_U$ is a Banach space, and $M$ believes that (2) is the dual of $(E_i)_U$, so $M$ will satisfy $\phi((E_i)_U,(2))$. This may be enough for some purposes, or it may even imply that $\phi((E_i)_U,(2))$ holds in "the real world".)
