Proposition. Let and let be a linear (respectively, nonnegative sublinear) operator defined
on a dense linear subspace of and taking values in . If
for all , then has a unique extension to a linear operator from to for which the inequality holds for all .
Proof. First, we assume that is linear. Fix . Since is dense in , there is a sequence in such that as . Then note that by linearlity,
holds for any by assumption. Since is a Cauchy sequence in , is a Cauchy sequence in . So by completeness of , converges in , say .
Next, the limit is independent of the approximating sequence. Let be another sequence in that converges to in . Then
and since , we conclude that converges to a limit in that equals to the limit of .
Finally, if and in , respectively, then and .
holds for all .
Next, assume is nonnegative sublinear operator. Note that
Similarly, . Hence
The rest part are similar.