For Proposition 2.5 in Lang's Fundamentals of Differential Geometry, it is stated

Let $E,F$ be Banach spaces. Then the set of toplinear (that is, bounded linear) isomorphisms Lis($E,F$) is open in $L(E,F)$ (the set of bounded linear maps)

And the proof is as follows:

If Lis($E,F$) is not empty, one is immediately reduced to proving that Laut($E$) is open in L($E,E$). We then remark that if $u \in L(E,E)$ and $|u| < 1$, then the series $1 + u + u^2 + ...$ converges. Given any toplinear automorphism $w$ of $E$, we can find an open neighborhood by translating the open unit ball multiplicatively from 1 to $w$.

I am not able to follow this argument very well.

The reduction to the case of endomorphisms and automorphisms is believable enough, but after that, I cannot even tell how the following statements are related to each other, unfortunately.

It seems to me that they are indicating that for a bounded linear map $u$ of operator norm less than 1, that the operator $(1-u)^{-1} = \sum_{i \geq 0} u^i$ is well defined.

However, I have never heard the phrase "translate the open unit ball multiplicatively" before, and it is unclear to me what is meant.

For Proposition 2.5 of Chapter I in Lang's Fundamentals of Differential Geometry, it is stated

Let $E$, $F$ be Banach spaces. Then the set of toplinear (that is, bounded linear) isomorphisms ${\rm Lis}(E,F)$ is open in $L(E,F)$ (the set of bounded linear maps).

And the proof is as follows:

If Lis($E,F$) is not empty, one is immediately reduced to proving that ${\rm Laut}(E)$ is open in $L(E,E)$. We then remark that if $u \in L(E,E)$ and $|u| < 1$, then the series \begin{equation} 1 + u + u^2 + \ldots \end{equation} converges. Given any toplinear automorphism $w$ of $E$, we can find an open neighborhood by translating the open unit ball multiplicatively from $1$ to $w$.

I am not able to follow this argument very well.

The reduction to the case of endomorphisms and automorphisms is believable enough, but after that, I cannot even tell how the following statements are related to each other, unfortunately.

It seems to me that they are indicating that for a bounded linear map $u$ of operator norm less than 1, that the operator $(1-u)^{-1} = \sum_{i \geq 0} u^i$ is well defined.

However, I have never heard the phrase "translate the open unit ball multiplicatively" before, and it is unclear to me what is meant.