Probabilistic normed linear spaces (briefly PNL spaces) were first studied by A. N. Serstnev in [1]. His definition was motivated by the definition of probabilistic metric spaces (PM spaces) which were introduced by K. Menger and subsequebtly developed by A. Wald, B. Schweizer, A. Sklar and others.
In a previuos paper [2] we studied the relationship between two important classes of PM spaces, namely E-spaces and pseudo-metrically generated PM spaces. We showed that a PM space is pseudo-metrically generated if and only if it is isometric to an E-space. In this paper we extend these ideas and results to PNL spaces. We define E-norm spaces and pseudo-norm-generated PNL spaces and show that a PNL space is pseudo-norm-generated if and only if it is isomorphically isometric to an E-norm space. In order to preserve the algebraic structure in a meaningful way we require new constructions which differ considerably from those given in the original PM space setting.
@article{urn:eudml:doc:38819,
title = {Isomorphically isometric probabilistic normed linear spaces.},
journal = {Stochastica},
volume = {3},
year = {1979},
pages = {71-77},
zbl = {0432.60014},
mrnumber = {MR0556648},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38819}
}
Sherwood, Howard. Isomorphically isometric probabilistic normed linear spaces.. Stochastica, Tome 3 (1979) pp. 71-77. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38819/