Sentence examples for functional on for from inspiring English sources

We now define the functional on, for, (4.31).

Then, is a nonnegative continuous concave functional on and for each.

Then it is easy to check that is a nonnegative continuous concave functional on with for and is completely continuous.

Suppose that is a completely continuous operator and is a nonnegative continuous concave functional on with for all.

Since tr is a positive linear functional on and for every we have ; therefore is a (semi-Hilbert) pre-Hilbert -module.  .

Since tr is a positive linear functional on and for every we have ; therefore is a (semi-Hilbert) pre-Hilbert -module.

Let be a completely continuous operator and let be a nonnegative continuous concave functional on such that for all.

Let be completely continuous, and let be a nonnegative continuous concave functional on such that for all.

Let be a completely continuous operator and a nonnegative continuous concave functional on such that for all.

Assume that and are nonnegative continuous convex functionals satisfying (H1) and (H2), is a nonnegative continuous concave functional on such that for all, and is a complete continuous operator.

Assume that and are nonnegative continuous convex functionals satisfying (2.6) and (2.7), is a nonnegative continuous concave functional on such that for all, and is a completely continuous operator.