= Duality mapping
{title2=$\mathcal J(x)=\partial\bigl(\tfrac12\|x\|^2\bigr)$}
The <duality mapping> takes a vector to its supporting dual functionals for half its squared <norm>. Equivalently $p\in\mathcal J(x)$ when $\|p\|_* =\|x\|$ and $\langle x,p\rangle=\|x\|^2$. This follows from the dual-norm inequality and equality in the conjugate relation for squared norms. In a <Hilbert space>, the <Riesz representation theorem> identifies it with the identity; in a general <Banach space> it can be set-valued.
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