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.


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


File Type





© 2023 American Mathematical Society, All rights reserved.

Publication Date

01 Jan 1976