Semi-Gaussian Subspaces

Abstract

Let M be a subspace (=closed linear manifold) of an L₂ space. Then M may or may not have the following property: If {x _k } is any orthogonal sequence in M such that $\sum\nolimits_{k = 1}^\infty {||{x_k}|{|^2} < \infty},$ then the series $\sum\nolimits_{k = 1}^\infty {{x_k}}$ 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.

Received by the editors August 18, 1961.