Group of invertible elements of a Banach algebra
= Group of invertible elements of a Banach algebra
{title2=$G(A)$}
The invertible elements of a unital <Banach algebra> form an open multiplicative <group>. For $a\in G(A)$, every $b$ satisfying $\|a^{-1}\|\|b-a\|<1$ is invertible, because $b=a(1+a^{-1}(b-a))$ and the second factor has a <Neumann series> inverse.