  Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of
n-dimensional vector spaces. However, while n-dimensional spaces in
applications are always realized as the Euclidean space R^n, Hilbert spaces
admit various useful realizations as spaces of functions. In the paper this
simple observation is used to construct a fruitful formalism of local
coordinates on Hilbert manifolds. Images of charts on manifolds in the
formalism are allowed to belong to arbitrary Hilbert spaces of functions
including spaces of generalized functions. Tensor equations then describe
families of functional equations on various spaces of functions. The formalism
itself and its applications in linear algebra, differential equations and
differential geometry are analyzed.
