# Definition:Functional/Weak Extremum

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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.

Suppose, there exists $\epsilon>0$ such that for $\size{y-\hat y}_1<\epsilon$ the expression $J\sqbrk y-J\sqbrk{\hat y}$ has the same sign for all $y$.

Here $\size{~}_1 $ denotes the norm of in the space $C^1$.

Then $y=\hat y$ is a **weak extremum** of the functional $ J\sqbrk y$.

## 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