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International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 14, Pages 2241-2275
http://dx.doi.org/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.