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

## Sources

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