Semi-Gaussian subspaces
D. L. Burkholder · Transactions of the American Mathematical Society · 1962
Introduction.Let M be a subspace (= closed linear manifold) of an L2 space.Then M may or may not have the following property: If {x*} is any orthogonal sequence in M such that ¿"2t°-i ||x*||2< °°, then the series ]dtli ** converges almost everywhere.That is, for orthogonal expansions in M, convergence in the mean implies convergence almost everywhere.If M does have this property, we say that M is semi-Gaussian.Clearly, Gaussian subspaces are semi-Gaussian.For if {x*} is an orthogonal sequence in a Gaussian subspace, then {x*j is a mutually independent random variable sequence-this property and the property that if x is in the subspace, then £x = 0, Ex2~^0, and the distribution of x is Gaussian, characterize Gaussian subspaces in both the real and the complex cases-and the mean convergence of the series z3T=i x* implies its convergence almost everywhere.Here, and throughout the paper, the same symbol may denote, depending on the context, either an equivalence class (of measurable functions equal almost everywhere) or one of its members.Any finite dimensional subspace is semi-Gaussian.So is any subspace of a semi-Gaussian subspace.Other examples may be obtained by using the methods of §4 or the result of §5.