Operator on $\ell^1(\mathbb{N})$ space

Question

Let $A = (A_{n,m})$ with $A_{n,m} \in \mathbb{C}, n,m \in \mathbb{N}$ be a matrix. Assume $\|A\| = \sup_m \sum_n|A_{n,m}| < \infty$. Show that for $T:\ell^1(\mathbb{N}) \to \ell^1(\mathbb{N}), (Tf)(n) = \sum_mA_{n,m}f(m)$ we have $\|T\| = \|A\|$.

Showing $\|T\| \leq \|A\|$ was easy. But how can I show $\|T\| \geq \|A\|$?

Answer 1

Let $e_k$ be the sequence with $1$ at the $k$-th place and $0$ everywhere else. Then $\|Te_k\|=\sum_n |T(e_k)(n)|= \sum _n |A_{n,k}|$ so $\|T\| \geq \sum _n |A_{n,k}|$ for all $k$. Take sup over $k$.