Potpourri, 8

for functions f on E. Let V be a finite-dimensional real or complex vector space. A linear transformation A from real or complex-valued functions on E into V , as appropriate, is characterized by the images under A of the functions on E which are equal to 1 at one element of E and equal to 0 elsewhere on E. A linear transformation T from V into real or complex-valued functions on E, as appropriate, is basically the same as a collection of linear functionals on V , one for each element of E, corresponding to evaluating images of vectors in V under T at elements of E. Suppose that ‖·‖V is a norm on V . Thus ‖v‖V is a nonnegative real number for all v ∈ V , ‖v‖V = 0 if and only if v = 0, ‖αv‖V = |α| ‖v‖V for all real or complex numbers α, as appropriate, and all v ∈ V , and