Exercise 7, § 2

Let $A$ be a normed algebra.

(a) Let $S$ be a bounded nonempty subset of $A$. Prove that the limit

$\varrho (S) =$ lim$_{n\rightarrow+\infty}$ sup$_{x\in S^n}\|x\|^{1/n}$

exists and that $\varrho (S) =\varrho (x)$ if $S$ is reduced to a single element $x$. The number $\varrho (S)$ is called the spectral radius of the subset $S$.

(b) Let $N$ be the set of norms $p$ on $A$, equivalent to the norm of $A$, and such that $A$, endowed with $p$, is a normed algebra. Let $X$ be a subset of $A$. In order that there exist $p\in N$ such that $p(x)\leqslant 1$ for every $x\in X$, it is necessary and sufficient that there exist a real number $c\geqslant 0$ such that

sup$_{n\geqslant 1}$ sup$_{x\in X^n}\|x\|\leqslant c$.

(c) Prove that

$\varrho (S) =$ inf$_{p\in}$sup$_{Nx\in S}p(x)$.