Definition:Sequence/Minimizing/Functional/Limit Minimizing Function of

Definition

Let $\sequence {y_n}$ be a minimizing sequence of a functional $J$.

Suppose:

$\displaystyle \lim_{n \mathop \to \infty} y_n = \hat y$

and

$\displaystyle \lim_{n \mathop \to \infty} J \sqbrk {y_n} = J \sqbrk {\hat y}$

Then $\hat y$ is the limit minimizing function of $J \sqbrk {y_n}$ and $J \sqbrk {\hat y} = \mu$.