Prove that if $E$ is Banach space then $\text {GL} (E)$ is an open subset of $\mathcal L (E).$
Here $\text {GL} (E)$ denotes the space of all bounded invertible linear operators on $E$ with bounded inverse and $\mathcal L (E)$ is the space of all bounded linear operators on $E.$
Let $T \in \text {GL} (E).$ We need to show that there exists an open ball $B(T, r)$ surrounding $T$ with radius $r \gt 0$ such that $B(T,r) \subseteq \text {GL} (E).$ Let $S \in B(T,r).$ If we can show that $\|I – S \|_{\text {op}} \lt 1$ then we are through. But how do I prove that? Any help will be appreciated.
Thanks in advance.
Best Answer
Hint: $S$ is necessarily invertible if $I - T^{-1}S = T^{-1}(T - S)$ has a sufficiently small operator norm.