Chapter 19

Show that \(f=g\) in \(\mathcal{F}\) if and only if \(f=g\) as linear functionals, which holds if and only if \(f=g\) as linear operators.

Prove that if \(T\) is an automorphism of the umbral algebra, then \(T\) preserves order, that is, \(o(T f(t))=o(f(t))\). In particular, \(T\) is continuous.