Suppose we have a Banach space ultraproduct $(E_i)_U$. I can think of three natural objects which are "dual-like":

1. $(E_i^*)_U$, the ultraproduct of the duals of the ground spaces.

2. 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 $x$, $|f_i(x)|\leq B||x_i||$

   c) For any $(x_i)_U, (y_i)_U\in (E_i)_U$, $\lim_U||f_i(x_i+y_i)-f(x_i)-f(y_i)||=0$

   d) For any $(x_i)_U\in (E_i)_U$ and any real $\alpha$, $\lim_U ||f_i(\alpha x_i)-\alpha f_i(x_i)||=0$

3. 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?