Sentence examples for be the pairing between from inspiring English sources

Let be a real Banach space with the topological dual space and let be the pairing between and.

Let C be a nonempty, closed, and convex subset of E and 〈 ⋅, ⋅ 〉 be the pairing between E and  E ∗.

Let E be a Banach space with dual E ∗ and let 〈 ⋅, ⋅ 〉 be the pairing between E and E ∗.

Let E be a real Banach space with the norm || · ||, E* be the dual of E, and 〈·,·〉 be the pairing between E and E*.

Let be a real Banach space with the topological dual space of, let be the pairing between and, let denote the family of all subsets of and let denote the family of all nonempty closed bounded subsets of.

Throughout this paper, we assume that E is a real reflexive Banach space with the dual space of (E^) and (langlecdot,cdotrangle) is the pairing between E and (E^).

Throughout the paper, unless otherwise specified, we assume that is a real Banach space with its norm,   is the topological dual of,   is the pairing between and,   is the metric induced by the norm,   (resp., ) is the family of nonempty (resp., nonempty closed and bounded) subsets of  ,  and is the Housdorff metric on defined by (21).

Here, u(r ij ) is the pair potential between particle i and particle j which are r ij apart.

Let 〈 ⋅, ⋅ 〉 be the duality pairing between L p ′ ( a, b ) and L p ( a, b ).

In Study 1, purple and orange versions of the test foods were formulated (the pairing between colour and energy density was counterbalanced across participants).

Linkage of the fields was achieved by identifying those which were to be paired between the two databases.