Continuous families of linear functionals

Abstract

We begin by recalling an elementary (but not widely advertised) theorem concerning finite families of linear functionals.After that, the principle aim is an analogous theorem (again not widely advertised) concerning suitable infinite continuous families of linear functionals.The discussion might well find a place in a first course on linear functional analysis, especially if it is desired to provide some concrete applications of general theorems. In all that follows E is assumed to be a complex linear space, f a linear functional on $E, T$ a set, and $(f_i)_{i∈ T}$ a family of liner functionals on E. Trivial changes take care of the case in which E is a real linear space.