Proposition 33.2.3.label Let $A$ be a unital Banach algebra, then:
- (1)
$G(A)$ is open.
- (2)
For any $x \in G(A)$ and $y \in B_{A}(0, \normn{x^{-1}}_{A}^{-1})$,
\[(x - y)^{-1}= x^{-1}\sum_{n = 0}^{\infty} (yx^{-1})^{n}\] - (3)
The map $G(A) \to G(A)$ defined by $x \mapsto x^{-1}$ is $C^{\infty}$.
Proof. (2): For any $x \in G(A)$ and $y \in B(0, \normn{x^{-1}}_{A}^{-1})$, $(x - y) = (1 - yx^{-1})x$. By Lemma 33.2.2,
\[(1 - yx^{-1})^{-1}= \sum_{n = 0}^{\infty} (yx^{-1})^{n}\]
so
\[(x - y)^{-1}= x^{-1}\sum_{n = 0}^{\infty} (yx^{-1})^{n}\]
(3): Since the inversion map is locally a power series, it is $C^{\infty}$ by Theorem 30.7.3.$\square$
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