Definition:Sequence/Minimizing/Functional

Definition
Let $ y $ be a real mapping defined on space $ \mathcal M $.

Let $ J \left [ { y } \right ] $ be a functional such that:


 * $ \displaystyle \exists y \in \mathcal M : J \left [ { y } \right ] < \infty $


 * $ \displaystyle \exists \mu > - \infty : \inf_y J \left [ { y } \right ] = \mu $

Let $ \left \{ { y_n } \right \} $ be a sequence such that:


 * $ \displaystyle \lim_{ n \to \infty } J \left [ { y_n } \right ] = \mu $

Then a sequence $ \left \{ { y_n } \right \} $ is called a minimizing sequence.