# Article

**Summary:**

[For the entire collection see Zbl 0699.00032.] \par The paper deals with a special problem of gauge theory. In his previous paper [The invariance of Sobolev spaces over noncompact manifolds, Partial differential equations, Proc. Symp., Holzhaus/GDR 1988, Teubner- Texte Math. 112, 73-107 (1989; Zbl 0681.58011)], the author introduced the Sobolev completions $\bar {\cal C}\sp k\sb P$ of the space ${\cal C}\sb P$ of all G-connections on a G-principal fibre bundle P. In the present paper, under the assumption of bounded curvatures and their covariant derivatives up to order k, the closedness of the subspace im $\nabla\sp{\omega}$ is proved to be a property of the whole component comp($\omega$) of a connection $\omega\in {\cal C}\sb P$ in the completion $\bar {\cal C}\sp k\sb P$. The result follows from the fact that the essential spectrum of the Laplacian $\Delta\sp{\omega}$ is the same for all $\omega$ lying in the mentioned component.