= Solution
On a real <Hilbert space>, for a proper lower-semicontinuous <convex function> $E$, the <proximal operator> is
$$
\boxed{\operatorname{prox}_E(q)=\arg\min_x\left\{E(x)+\frac12\|x-q\|^2\right\}.}
$$
The quadratic makes this objective <strongly convex> and coercive; the affine lower bound available for a proper closed <convex function> and <weak lower semicontinuity> give existence. The minimizer is unique. The <subdifferential> optimality condition is
$$
\boxed{q-x\in\partial E(x),\qquad x=\operatorname{prox}_E(q).}
$$
For a positive parameter $t$, $\operatorname{prox}_{tE}$ uses $tE(x)$ in the same formula. On a general <Banach space>, the analogous norm-squared minimization is <Banach-space proximal minimization>; it can be set-valued and the identity-valued Hilbert derivative must be replaced by the <duality mapping>.
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