A continuous basis for Orlicz spaces

Abstract

1. In 1910 Haar created the orthonormal system that bears his name in order to show that there are ON systems with respect to which the Fourier series of each continuous function converges uniformly to the function. The Haar functions are themselves discontinuous, however, and this led Franklin to construct a continuous ON set that plays the same role. Now the Haar functions comprise a Schauder basis for each of the spaces LP [0, 1], p _ 1, as Schauder himself has shown [5]. Indeed, this system serves as a Schauder basis for each of the separable Orlicz spaces associated with the unit interval, see [2], [4]. Thus, it is natural to inquire whether the Franklin system also serves as a basis for these spaces. In the present article this question is answered affirmatively. The proof is elementary, requiring only the judicious application of the uniform boundedness principle and the Jensen integral inequality. For the spaces LP, p _ 1, this result has been noted recently by Ciesielski [1]. Nevertheless, even for this special case, the demonstration given below may be of interest by virtue of its utter simplicity.