## Smooth partitions of unity over Banach manifolds.(English. Russian original)Zbl 0928.58011

Russ. Math. 41, No. 10, 49-55 (1997); translation from Izv. Vyssh. Uchebn. Zaved., Mat 1997, No. 10(425), 51-58 (1997).
The author separates out a class of Banach manifolds (in particular, Banach spaces) which admit a $$C^r$$-partition of unity $$(r\geq 2)$$ of a special type. The practical need to construct those special partitions of unity arises in the author’s attempt to transfer the theory, developed by Yu. G. Borisovich and herself, of boundary indices of sets of generalized eigenvectors of a pair of nonlinear operators in Hilbert spaces to Banach spaces. The well-known Eells-Lang’s theorem on the existence of a $$C^r$$-partition of unity for a paracompact Hausdorff $$C^r$$-manifold, where $$r \geq 1$$, which is modeled by a separable Hilbert space, is deduced here as a corollary of a more general theorem (Corollary 3.2 of Theorem 3.2).

### MSC:

 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 58C40 Spectral theory; eigenvalue problems on manifolds

### Keywords:

partitions of unity; Banach manifolds