Naming frustration - "dual space" vs "space of duals" are not the same

If we're in the 5-limit, the "dual space" to the space of bimonzos is
the space of bivals. But the "space of duals" to the space of bimonzos
is the space of vals.

The former space consists of linear functionals on bimonzos, aka
bivals; it is the "dual space". The latter space is the space which
consists of "the duals," in the xenwiki sense, to each bimonzo; it is
the "space of duals."

The term "dual space" clearly and unambiguously refers to the first
space; is there a better way for me to refer to the second space?
Could I call it the kernel space or something?

-Mike