Uniformly Bounded invertible linear operators have their inverses with norms uniformly bounded

Question

Let $\{T_n\}$ be a series of bounded linear operators on $X$ to $Y$, where $X,Y$ are Banach space. Suppose that $T_n$ is invertible, and $\sup_n ||T_n||<\infty$. Can we show that $\sup_n ||T_n^{-1}||<\infty$?

Answer 1

Think about the operators $T_n = 1/n$ on $X = Y = \mathbb{R}$.