Exercise 34, § 2

Let A be a unital algebra over $\mathbf{C}$, endowed with a separated locally convex topology such that the multiplication in A is separately continuous. An element $a\in A$ is said to be regular if there exists $r\geqslant 0$ such that $a-\lambda$ is invertible for $|\lambda |\geqslant r$, and such that the set of $(a-\lambda )^{-1}$, for $|\lambda |\geqslant r$, is bounded.

a) If $a$ is regular, $\lambda (a-\lambda )^{-1}$ remains bounded for $|\lambda |\geqslant r$.

b) If the map $x\mapsto x^{-1}$ is defined in a neighbourhood of 1 and continuous at the point 1, every element of A is regular.

c) For every $a\in A$, let $U_a$ be the set of $\lambda \in \mathbf{C}$ such that $(a-\mu)^{-1}$ exists and is bounded for all $\mu$ in a neighbourhood of $\lambda$. Let $S_a=\mathbf{C}-U_a$. Then $S_a$ is closed; and, if $a$ is regular, $S_a$ is compact nonempty (provided that $A\not=\{0\}$).

d) Let $a\in A$. The function $\lambda \mapsto (a-\lambda )^{-1}$ is holomorphic in $U_a$.

e) Let $a\in A$. For $\lambda \in U_a$, it is necessary and sufficient that $a-\lambda$ possesses a regular inverse.

f) If every element of A is regular, and if A is a field, one has $A =\mathbf{C}\cdot 1$.