A class of functional equations and Mielnik probability spaces

Authors:
S. J. Guccione and Č. V. Stanojević

Journal:
Proc. Amer. Math. Soc. **59** (1976), 317-320

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 two-dimensional 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.

**[1]**C. V. Stanojevic,*Mielnik’s probability spaces and characterization of inner product spaces*, Trans. Amer. Math. Soc.**183**(1973), 441–448. MR**0328562**, 10.1090/S0002-9947-1973-0328562-1**[2]**Bogdan Mielnik,*Geometry of quantum states*, Comm. Math. Phys.**9**(1968), 55–80. MR**0231603****[3]**P. L. Ganguli,*A note on the integral of some classes of functions*, Bull. Calcutta Math. Soc.**57**(1965), 63–68. MR**0210848****[4]**A. R. Blass and C. V. Stanojevic,*Partial Mielnik spaces and characterization of uniformly convex spaces*, Proc. Amer. Math. Soc.**55**(1976), no. 1, 75–82. MR**0394121**, 10.1090/S0002-9939-1976-0394121-6

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1976-0454605-9

Keywords:
Mielnik probability spaces,
functional equation

Article copyright:
© Copyright 1976
American Mathematical Society