# Definition:Sequence/Minimizing/Functional

## 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**.

## Sources

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