Sentence examples for totally bounded from inspiring English sources

It is shown constructively that if the unit ball of R is weak-operator totally bounded, then an ultraweakly continuous linear functional on R extends to one on B(H), and the extended functional has the form T↦∑∞n=1 〈Txn, yn〉, where ∑∞n=1 ‖xn‖2 and ∑∞n=1 ‖yn‖2 are convergent series in R.

Then X is totally bounded.

This implies that N is totally bounded.

Therefore, the set is totally bounded.

Suppose such that is totally bounded.

Since the ranges and are totally bounded, we know that the range is also totally bounded in ; that is, is totally bounded in.

We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (see [[27], Theorem 2.10]).

We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (cf. [22]).

It is also well known that g is totally convex on bounded sets if and only if the function g is uniformly convex on bounded sets (see [[41], Theorem 2.10, p.9]).

Since f is totally convex on bounded subsets of E, f is sequentially consistent (see [35]).

g is sequentially consistent if and only if it is totally convex on bounded sets.