Sentence examples for be the dual operator from inspiring English sources
Let (T_{beta}') be the dual operator of (T_{beta}).
end{aligned} (4.5 Let (mathcal {A}) be the dual operator to A determined by begin{aligned} _{mathfrak {H}_{-2}}langle mathcal {A}f,grangle _{mathfrak {H}_2}:= f,Ag)_{mathfrak {H}}, quad fin mathfrak {H}, quad gin mathfrak {H}_2.
Let (Upsilon _j^*:mathcal {H}rightarrow mathfrak {H}_{-2}) be the dual operator to (Upsilon _j:mathfrak {H}_2rightarrow mathcal {H}), (j=0,1), in (4.7) determined by begin{aligned} _{mathfrak {H}_{-2}}langle Upsilon _j^* varphi,grangle _{mathfrak {H}_2}:= varphi,Upsilon _j g)_{mathcal {H}}, quad varphi in mathcal {H}, ; gin mathfrak {H}_2, ; j=0,1.
Since the operator (mathcal{H}_{varphi,m}^{p,*}) is the dual operator of (mathcal{H}_{varphi,m}^{p}), we can immediately deduce the following result.
Let be a real Banach space and let be the dual space of Let be a maximal monotone operator from to.
If the dual operator L ∗ is monotone, then L is maximal monotone.
Moreover, by Lemma 3.5 the dual operator P q ∗ is a bounded linear in L q ′ ( G ; E ∗ ).
Moreover, by Lemma 4.4, the dual operator P q ∗ is bounded linear in L q ′ ( R n ; E ∗ ).
Moreover, in Section 5 some similar results for the dual operator (T_{alpha, beta}^{0}) are given.
