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Consider the following theorem in functional analysis. Let $X$ be a Banach space. Suppose that $S,T \in \mathscr{B}(X)$, $T$ is invertible, and  $||T-S||<||T^{-1}||^{-1}$. Then $S$ is invertible in $ \mathscr{B}(X)$. We see two notes about this theorem.   NOTE 1: At first glance, one may ask that   "Can we write$ ||T^{-1}||^{-1}$ as ||T||?". Because $TT^{-1}=I$, $||I||=1$ and "norm is generalization of modulus function |.| for which we have $|AB|=|A||B|$." So it is natural to have such a thought. But it is interesting to note that  $||AB|| \neq ||A||||B||$, but $||AB|| \leq ||A||||B||$. For example consider the matrix  $\begin{bmatrix}0&\frac{1}{2}\\2&0 \end{bmatrix}$. then 𝑀 − 1 = 𝑀 . Note that  ‖ 𝑀 ‖ ≥ ‖ 𝑀 𝑒 1 ‖ / ‖ 𝑒 1 ‖ = 2 . Hence ‖ 𝑀 − 1 ‖ = ‖ 𝑀 ‖ ≥ 2 .  Thus the equality ‖ 𝑀 − 1 ‖ = ‖ 𝑀 ‖ − 1 doesn't hold in general. Anyway, in general we can have ‖ 𝑇 − 1 ‖ ≥ 1 ‖ 𝑇 ‖ NOTE 2: T