#### Abstract

Let S be the unit sphere of a normed real linear space N and let (S, p) be a Mielnik space of dimension two. For p(x, y) = f(‖x+y‖), x, yєS, where /is a continuous, strictly increasing function from [0, 2] onto [0, 1], it has been shown that (S, p) being two dimensional is equivalent to N being an inner product space. In some polarization problems modeled on the unit sphere of an inner product space, the transition probability p(x, y) may not be as well behaved as p(x, y) = f(‖x + y‖). In order to provide a more suitable setting, we have constructed wide classes of two-dimensional transitional probability spaces (S, p), all having the same set of bases ℬ with p=⌽ ° f where ⌽ is a solution of a certain functional equation. In particular, for p(x, y) = ‖x+y‖24, we answer a question due to B. Mielnik. © American Mathematical Society 1976.

#### Recommended Citation

S.
J.
Guccione
and
C.
V.
Stanojevic,
"A Class Of Functional Equations And Mielnik Probability Spaces," *Proceedings of the American Mathematical Society*, vol. 59, no. 2, pp. 317 - 320, American Mathematical Society, Jan 1976.

The definitive version is available at https://doi.org/10.1090/S0002-9939-1976-0454605-9

#### Department(s)

Mathematics and Statistics

#### Keywords and Phrases

Functional equation; Mielnik probability spaces

#### International Standard Serial Number (ISSN)

1088-6826; 0002-9939

#### Document Type

Article - Journal

#### Document Version

Citation

#### File Type

text

#### Language(s)

English

#### Rights

© 2023 American Mathematical Society, All rights reserved.

#### Publication Date

01 Jan 1976