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

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Definition

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

Suppose:

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

and

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

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

This article, or a section of it, needs explaining. In particular: Ambiguous. Does the "and" separate two distinct clauses, or two elements of a list of elements that $\hat y$ is the limit minimizing function of, the second element just happening to equal $\mu$? If the former, then that second clause needs to be separated off into a separate page, and if the latter, make it more obvious that this is a two-element list by presenting it as one. It is the former. Will see how to reformulate it. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 8.39$: Minimizing Sequences