The predual "M"∗ is a closed subspace of the dual "M*" (which consists of all norm-continuous linear functionals on "M") but is generally smaller. The proof that "M"∗ is (usually) not the same as "M*" is nonconstructive and uses the axiom of choice in an essential way; it is very hard to exhibit explicit elements of "M*" that are not in "M"∗. For example, exotic positive linear forms on the von Neumann algebra "l"∞("Z") are given by free ultrafilters; they correspond to exotic *-homomorphisms into "C" and describe the Stone–Čech compactification of "Z".