Proposition 12

Let A be a commutative unital Banach algebra. The image of the exponential mapping is the identity component of the group G of invertible elements of A.

Formulas (10) and (11) prove that exp(A) is a subgroup of G. By what precedes (see formula (12)), this subgroup contains the open ball with center 1 and radius 1. It is therefore an open subgroup, and hence a closed subgroup, of G. On the other hand, A is connected and the mapping $a\mapsto$ exp($a$) is continuous, so that exp(A) is connected. Therefore exp(A) is the identity component of G.