International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 14, Pages 2241-2275
doi:10.1155/IJMMS.2005.2241
Linear algebra and differential geometry on abstract Hilbert space
Department of Mathematics, University of Wisconsin Colleges, 780 Regent Street, Madison 53708, WI, USA
Received 23 January 2003; Revised 17 June 2005
Copyright © 2005 Alexey A. Kryukov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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 Rn, 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 carefully analyzed.