Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications:
Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded linear operator $T \colon X \to Y$, where $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$) and $R(T)$ denotes the range of $T$, need not be bounded.
As a hint, Kryszeg suggest the following operator:
Let $T \colon \ell^\infty \to \ell^\infty$ be defined by
$$ Tx \colon = \left(\frac{\xi_j}{j}\right)_{j=1}^\infty \, \, \, \forall x \colon= (\xi_j)_{j=1}^\infty \in \ell^\infty. $$
How to characterise the range of this operator?
And how to show that the inverse of this operator is not bounded?
Best Answer
The range of this operator is a subspace of $C_{0}$, which consisting of elements eventually go to zero. It can be characterized by $$ \{a_{i}\}\in l^{\infty}, \exists N\in \mathbb{N}, |a_{i}*i|\le C, \forall i\ge N $$ It is not difficult to see that the inverse map $$ \{a_{i}\}\rightarrow \{ia_{i}\} $$ is not bounded on the sequence $a_{i}=1,\forall i$ under $l^{\infty}$-norm.