I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits.
Wiki gives a link to Toeplitz original paper yet it was written in a language that I cannot read. There is a proof on PlanetMath yet it only contains the proof of sufficiency not the necessity.
Can someone here point me to the proof of the necessity part? Or just give a hint on how to prove this? I got stuck on showing the uniform boundedness of the sums of each row of the matrix.
Thanks!
More or less straight from Toeplitz's paper:
To prove necessity of the third condition that $$\exists M \, \forall p: \; Z_p := \sum_q |a_{pq}| \le M,$$ we construct a sequence $s_1,s_2,...$ that tends to $0$, where $t_1 = a_{11} s_1 + a_{1n_1}s_{n_1}$, $t_2 = a_{21} s_2 + ... + a_{2n_2} s_{n_2}$, ... gives a divergent sequence $t_1,t_2,...$ under the assumption that $(Z_p)$ is unbounded.
Choose $p'$ large enough that $Z_{p'} > 10^2$ and set all $s_i$ that appear in the expression for $t_{p'}$ to $\pm \frac{1}{10}$, having the same sign as the coefficient $a_{p'i}$. Then $$t_{p'} = \frac{1}{10} \sum_q |a_{p'q}| > \frac{10^2}{10} = 10.$$
Since we also assume $\lim_{p \rightarrow \infty} a_{pq} = 0$ for all $q$, we have for all large enough $p > p_0$ that $$|a_{p1} + ... + a_{pn_{p'}}| < 1,$$ and therefore $$|a_{p1}s_1| + ... + |a_{pn_{p'}}s_{n_{p'}}| < \frac{1}{10}.$$
Now choose $p'' > p_0$ such that $Z_{p''} > 10^4 + 10 + 1$, and define $$s_{n_{p'} + 1},...,s_{n_{p''}} = \pm \frac{1}{100},$$ $s_i$ having the same sign as the coefficient $a_i$. Then $$t_{p''} = \sum_{q=1}^{n_{p'}} (a_{p''q} s_q) + \frac{1}{100} \sum_{q=n_{p'} + 1}^{\infty} |a_{p''q}|$$ $$= \frac{1}{100} \sum_{q=1}^{\infty} |a_{p''q}| - \frac{1}{100} \sum_{q=1}^{n_{p'}} |a_{p''q}| + \sum_{q=1}^{n_{p'}} a_{p''q}s_q$$ $$\ge \frac{10^4 + 11}{100} - \frac{1}{100} - \frac{1}{10} = 10^2.$$ By repeating this pattern we can construct a sequence $s_n \rightarrow 0$ such that $t_n \rightarrow \infty$.