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Isomorphisms of separable Hilbert spaces are analogous to isomorphisms ofn-dimensional vector spaces. However, while n-dimensional spaces inapplications are always realized as the Euclidean space R^n, Hilbert spacesadmit various useful realizations as spaces of functions. In the paper thissimple observation is used to construct a fruitful formalism of localcoordinates on Hilbert manifolds. Images of charts on manifolds in theformalism are allowed to belong to arbitrary Hilbert spaces of functionsincluding spaces of generalized functions. Tensor equations then describefamilies of functional equations on various spaces of functions. The formalismitself and its applications in linear algebra, differential equations anddifferential geometry are analyzed.

Linear algebra and differential geometry on abstract Hilbert space