Definition:Sequence/Minimizing/Functional

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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 $J \sqbrk y$).


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


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