# 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:

$\exists y \in \MM: J \sqbrk y < \infty$
$\ds \exists \mu > -\infty: \inf_y J \sqbrk y = \mu$

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

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

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

### Limit Minimizing Function of Functional

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 $J \sqbrk {\hat y} = \mu$.