A class of functional equations and Mielnik probability spaces
Authors:
S. J. Guccione and Č. V. Stanojević
Journal:
Proc. Amer. Math. Soc. 59 (1976), 317320
MSC:
Primary 46C10
MathSciNet review:
0454605
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Abstract: Let be the unit sphere of a normed real linear space and let be a Mielnik space of dimension two. For , where is a continuous, strictly increasing function from onto , it has been shown that being two dimensional is equivalent to being an inner product space. In some polarization problems modeled on the unit sphere of an inner product space, the transition probability may not be as well behaved as . In order to provide a more suitable setting, we have constructed wide classes of twodimensional transitional probability spaces , all having the same set of bases , with where is a solution of a certain functional equation. In particular, for , we answer a question due to B. Mielnik.
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 Č. V. Stanojević, Mielnik probability spaces and characterization of inner product spaces, Trans. Amer. Math. Soc. 183 (1973), 441448. MR 48 #6904. MR 0328562 (48:6904)
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 B. Mielnik, Geometry of quantum states, Comm. Math. Phys. 9 (1968), 5580. MR 37 #7156. MR 0231603 (37:7156)
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 P. L. Ganguli, A note on the integral of some classes of functions, Bull. Calcutta Math. Soc. 57 (1965), 6368. MR 35 #1733. MR 0210848 (35:1733)
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 A. R. Blass and Č. V. Stanojević, Partial Mielnik spaces and characterization of uniformly convex spaces, Proc. Amer. Math. Soc. 55 (1976), 7582. MR 0394121 (52:14926)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197604546059
PII:
S 00029939(1976)04546059
Keywords:
Mielnik probability spaces,
functional equation
Article copyright:
© Copyright 1976
American Mathematical Society
