= Solution
Let $X=c_{00}$, the finitely supported sequences, with the $\ell^2$ norm. This normed space is not complete. Let $Y=\ell^2$ and define
$$
S(x_1,x_2,\ldots)=
\left(x_1,\frac{x_2}{2},\frac{x_3}{3},\ldots\right),
\qquad
T(x)=x.
$$
Both maps $X\to Y$ are bounded, and $S$ is injective. Moreover
$$
\operatorname{im}S=c_{00}=\operatorname{im}T,
$$
because multiplication of a finitely supported sequence by $(1,2,3,\ldots)$ preserves finite support. But
$$
(S^{-1}T)e_n=ne_n,
$$
so $\|(S^{-1}T)e_n\|_2=n$ while $\|e_n\|_2=1$. Hence $S^{-1}T$ is not continuous, showing that completeness of $X$ is essential.
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