Sequences of independent functions and structure of rearrangement invariant spaces

Abstract

The main aim of the survey is to present results of the last decade on the description of subspaces spanned by independent functions in L<inf>p</inf>-spaces and Orlicz spaces on the one hand, and in general rearrangement invariant spaces on the other. A new approach is proposed, which is based on a combination of results in the theory of rearrangement invariant spaces, methods of the interpolation theory of operators, and some probabilistic ideas. The problem of the uniqueness of the distribution of a function such that a sequence of its independent copies spans a given subspace is considered. A general principle is established for the comparison of the complementability of subspaces spanned by a sequence of independent functions in a rearrangement invariant space on [0, 1] and by pairwise disjoint copies of these functions in a certain space on the half-line (0, ∞). As a consequence of this principle we obtain, in particular, the classical Dor–Starbird theorem on the complementability of subspaces spanned by independent functions in the L<inf>p</inf>-spaces. © 2024 Elsevier B.V., All rights reserved.