Exercise 2, § 1

Let E be the vector space over C generated by the functions $x^n \ (n \in \mathbf{N})$, $e^{\lambda x} \ (\lambda \in \mathbf{C}, \lambda \neq 0)$ and $|x+\mu| \ (\mu \in \mathbf{R})$.

a) Show that the preceding functions form a basis for E.

b) Let U be the composition operator defined on E by the conditions: $U_0^\xi(\xi^n) = (n!)^2$, $U(e^{\lambda x}) = e^{\lambda x}$ for $\lambda \neq 0$, $U(|x+\mu|) = |x+\mu|$ for $\mu \in \mathbf{R}$. The indicatrix $u(\lambda)$ is the constant 1 but the series with general term $n! \lambda^n$ does not converge for any value $\lambda \neq 0$.

c) Let V be the composition operator defined on E by the conditions $V(x^n) = x^n$, $V(|x+\mu|) = |x+\mu|$, $V(e^{\lambda x}) = 0$ for $\lambda \neq 0$; the indicatrix $v(\lambda)$ is equal to 1 for $\lambda = 0$, to 0 for $\lambda \neq 0$, so is different from the sum of the series $\sum_{n=0}^\infty \alpha_n \lambda^n$ where $\alpha_n = V_0^\xi(\xi^n)/n!$.

d) Let W be the composition operator defined on E by

show that $VW \neq WV$.