# Definition:Sequence/Minimizing/Functional

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## Definition

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

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

$\exists y \in \mathcal M: J \sqbrk y < \infty$
$\displaystyle \exists \mu > -\infty: \inf_y J \sqbrk y = \mu$

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.