As a result, the set Uy(t) is also

relatively compact set in X for t [member of] [0, [infinity]).

Recall from [3] that an operator T : E [right arrow] X is said to be b-AM-compact if it carries b-order bounded set of E (i.e., order bounded in E") into norm relatively compact set of X.

Note that every relatively compact set is limited but the converse is not true in general.

Consider [mathematical expression not reproducible] the subfamily of [W.sup.n,1]([[0, T].sup.N]) consisting of

relatively compact sets in the topology [[tau].sup.[omega]] and [mathematical expression not reproducible] the family of all nonempty and bounded subsets (innorm) of [W.sup.n,1]([[0, T].sup.N]).

Therefore, letting [epsilon] [right arrow] 0, we see that, there are

relatively compact sets arbitrarily close to the set {[phi](t): [phi] [member of] B([B.sub.l])[parallel].

Because a well-known result states that in complete metric spaces,

relatively compact sets coincide with pre-compact sets, it is sufficient to show that for any [epsilon] > 0, the set of values of the function can be embedded in a finite number of spheres of radius [epsilon].