Operators on Complex Vector Spaces — Sheldon Axler
Mathematics — Linear Algebra · operator-norms-invertibility
Problem
(a) Prove that if T∈L(V) and ∥I−T∥<1, then T is invertible.
(b) Suppose that S∈L(V) is invertible. Prove that if T∈L(V) and ∥S−T∥<1/∥S−1∥, then T is invertible.
This exercise shows that the set of invertible operators in L(V) is an open subset of L(V), using the metric defined in Exercise 7.