Basic sequences in the space of measurable functions

Abstract

1. In a topological vector space X, a basic sequence [xn] is one whose finite linear combinations are dense in X. In a recent work, [l], A. A. Talalyan has observed that the space of measurable functions has a distinctly different character, with respect to the behavior of basic sequences, from, for example, the Lp spaces, p^l. A striking result of Talalyan is the fact that if { „} is basic, i.e., for every measurable , there are finite linear combinations of the , then if any finite number of functions is deleted from { „}, the remaining sequence is basic. This readily implies the existence of universal expansions, and the existence of a subsequence \<pnk which is basic even though the complement of the sequence {«*} is infinite. The proof given by Talalyan necessitates the use of considerable machinery from the theory of orthonormal systems in L2, and is quite involved. Our purpose is to show that the result follows almost immediately from the fact that the space M of measurable functions, with the topology of convergence in measure, has a trivial dual.