# Definition:Minimum Value of Functional

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## Definition

Let $y, \hat y \in S: \R \to \R$ be real functions.

Let $J \sqbrk y: S \to \R$ be a functional.

Let $J$ have a (relative) extremum for $y = \hat y$.

Let $J \sqbrk y - J \sqbrk {\hat y} \ge 0$ in the neighbourhood of $y = \hat y$.

Then this extremum is called the **minimum** of the functional $J$.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 1.3$: The Variation of a Functional. A Necessary Condition for an Extremum