Retractions onto the space of continuous divergence-free vector fields (Q661137)

In this paper, it is proved that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. The proof relies on the theory of absolutely summing operators and \(\mathcal{L}_{1}\) spaces. In the final section, this result is extended to a more general setting, using the concept of \(m\)-charges due to \textit{T. de Pauw}, \textit{L. Moonens} and \textit{W. F. Pfeffer} [J. Math. Pures Appl. (9) 92, No.~1, 86--112 (2009; Zbl 1166.49039)].