= Solution
\b[Compact.] Its column indexed by $m$ has entries $n!/m!$ for $1\leq n<m$, and zero otherwise. The sum of that column is $c_m=(1!+\cdots+(m-1)!)/m!\leq2/m$ for $m\geq2$, by the elementary induction $1!+\cdots+k!\leq2k!$. For operators from $\ell^1$ the supremum of the absolute column sums is the <operator norm>: one inequality is the <triangle inequality>, and coordinate unit vectors give the reverse. Keeping just the first $N$ input columns is finite rank and its remainder <norm> is at most $\sup_{m>N}c_m\leq2/(N+1)$. Hence it converges to $T$ in <norm>, proving compactness.
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