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Topologies on Generalized Inner Product Spaces

Published online by Cambridge University Press:  20 November 2018

Eduard Prugovečki
Eduard Prugovečki*
Affiliation:
University of Toronto, Toronto, Ontario
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In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.

The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 158 - 169
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bohm, A., Rigged Hilbert space and mathematical description of physical systems. Lectures in Theoretical Physics, Vol. 9A, edited by Brittin, W. B., Barut, A. O., and Guenin, M. (Gordon and Breach, New York, 1967).Google Scholar
2. Dirac, P. A. M., The principles of quantum mechanics (Oxford Univ. Press, Oxford, 1930).Google Scholar
3. Gel'fand, I. M. and Shilov, G. E., Generalized functions, Vol. 1 (Academic Press, New York, 1964).Google Scholar
4. Robertson, A. P. and Robertson, W., Topological vector spaces (Cambridge Univ. Press, Cambridge, 1964).Google Scholar
5. L., Schwartz, Généralisation de la notion de fonction, de dérivation, de transformation de Fourier, et applications mathématiques et physiques, Ann. Univ. Grenoble Sect. Sci. Math. Phys. (N.S.) 21 (1945), 57–74 (1946).Google Scholar