Continuous linear functionals on certain topological vector spaces

Abstract

Let X be a Lebesgue measurable subset of the real line and let L be the set of all real valued measurable functions on X, two functions being identified if they are equal almost everywhere. S. Banach and M. M. Day have provided the following real topological vector spaces which have no nonzero continuous linear functionals (see [1] EXAMPLE OF BANACH. Let V consist of all functions f in L for which \\f\ (1 + I/I)"