Coordinate formalism on abstract Hilbert space

Kryukov, Alexey (2001) Coordinate formalism on abstract Hilbert space.

Abstract

Coordinate formalism on Hilbert manifolds developed in \cite{Kryukov} is reviewed.
The results of \cite{Kryukov} are applied to the simpliest case of a Hilbert manifold:
the abstract Hilbert space. In particular, functional transformations preserving properties
of various linear operators on Hilbert spaces are found. Any generalized solution of an
arbitrary linear differential equation with constant coefficients is shown to be related
to a regular solution by a (functional) coordinate transformation.
The results also suggest a way of using generalized functions to solve nonlinear
problems on Hilbert spaces.