Definition:Weak Extremum

Definition
The functional $J[y]$ has a weak extremum for $y=\hat{y}$ if there exists $\epsilon >0$ such that $J[y]-J[\hat{y}]$ has the same sign for all $y$ in the domain of definition of the functional, which satisfy the condition $\left\Vert y-\hat{y}\right\Vert_1<\epsilon$ where $\left\Vert~\right\Vert_1$ denotes the norm of in the space $\mathscr{D}_1$.