On P. Levy's Stable Laws and Reflexive Subspaces of the Banach space L   of Lebesgue summable functions on [0,1]
  To describe a set of functions, which forms a reflexive subspace B of the classical Banach space L a special function that characterizes their average integral growth is introduced. It is shown that this function essentially depends on the geometry of B. By the way, one question of la Vallee Poussin is answered. Also a short proof of the known result about the existence of an uncomplemented subspace isomorphic to the Hilbert space in every Lebesgue - Riesz space Lp (1<p<2) is obtained. 
Eugene Tokarev