Let \(G\) be a simple algebraic group with Lie algebra \(\mathfrak g\). In this paper the authors prove a \(G\)-equivariant version of the Poincaré-Birkhoff-Witt theorem. In positive characteristic they also obtain analogous results for the hyperalgebras of \(G\) and of the Frobenius kernels \(G_r\), \(r\geq 1\). Extending \textit{F. D. Veldkamp}'s theorem [ibid. 5, 217--240 (1972; Zbl 0242.17009)], they are then able to identify the \(G_r\)-invariants both in the symmetric algebra and in the universal enveloping algebra of \(\mathfrak g\).