Definition:Sequence/Minimizing/Functional

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

Let $J \sqbrk y$ be a functional such that:


 * $\exists y \in \MM: J \sqbrk y < \infty$


 * $\displaystyle \exists \mu > -\infty: \inf_y J \sqbrk y = \mu$

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


 * $\displaystyle \lim_{n \mathop \to \infty} J \sqbrk {y_n} = \mu$

Then the sequence $\sequence {y_n}$ is called a minimizing sequence (of the functional $J \sqbrk y$).