The Set of Invertible Operators Is Open
A corollary is that the set of invertible operators between two Banach spaces B and B' is open in the topology induced by the operator norm. Indeed, let S : B → B' be an invertible operator and let T: B → B' be another operator. If |S – T | < |S–1|–1, then T is also invertible. This follows by writing T as
and applying the result in the previous section on the second factor. The norm of T–1 can be bounded by
Read more about this topic: Neumann Series
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