= Solution
Let $(T_m)$ be Cauchy in the <operator norm> on $\mathcal B(X,Y)$. For each $x\in X$,
$$
\lVert T_mx-T_lx\rVert
\leq\lVert T_m-T_l\rVert\,\lVert x\rVert,
$$
so $(T_mx)$ is Cauchy in the <Banach space> $Y$. Define
$$
Tx=\lim_{m\to\infty}T_mx.
$$
Pointwise passage to the limit shows that $T$ is linear. A norm-Cauchy sequence is bounded, say $\lVert T_m\rVert\leq M$, and hence
$$
\lVert Tx\rVert\leq M\lVert x\rVert.
$$
Thus $T$ is bounded. Finally, for fixed $m$,
$$
\lVert(T_m-T)x\rVert
=\lim_{l\to\infty}\lVert(T_m-T_l)x\rVert
\leq\left(\limsup_{l\to\infty}\lVert T_m-T_l\rVert\right)\lVert x\rVert.
$$
The right-hand coefficient tends to zero with $m$, so $T_m\to T$ in operator norm. Therefore
$$
\boxed{\mathcal B(X,Y)\text{ is a Banach space}.}
$$
This is the <Banach space of bounded linear operators>.
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