Coordinate formalism on Hilbert manifolds

The formalism of local coordinates on infinite-dimensional Hilbert manifolds is introduced. Images of charts on the manifolds are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. The corresponding local coordinate form of algebra of tensor fields on Hilbert manifolds is constructed. A single tensor equation in the formalism is shown to produce a family of functional equations on different spaces of functions. This allows for a “covariant” approach to the theory of generalized functions and suggests a way of using generalized functions in solving linear and nonlinear problems. Examples in linear algebra, differential equations, differential geometry and variational calculus are used to illustrate the results.