We design an operator from the infinite-dimensional Sobolev space \(\varvec{H}(\textrm{curl})\) to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: (1) it is defined over the entire \(\varvec{H}(\textrm{curl})\) , including boundary conditions imposed on a part of the boundary; (2) it is defined locally in a neighborhood of each mesh element; (3) it is based on simple piecewise polynomial projections; (4) it is stable in the \(\varvec{L}^2\) -norm, up to data oscillation; (5) it has optimal (local-best) approximation properties; (6) it satisfies the commuting property with its sibling operator on \(\varvec{H}(\textrm{div})\) ; (7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the \(\varvec{H}(\textrm{curl})\) space. We in particular employ it here to establish the two following results: (i) equivalence of global-best, tangential-trace- and curl-constrained, and local-best, unconstrained approximations in \(\varvec{H}(\textrm{curl})\) including data oscillation terms; and (ii) fully h- and p- (mesh-size- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise. As a result of independent interest, we also prove a p-robust equivalence of curl-constrained and unconstrained best-approximations on a single tetrahedron in the \(\varvec{H}(\textrm{curl})\) -setting, including hp data oscillation terms.
