A point about \({\mathbb{N}}\times {\mathbb{N}}\) matrices and \(\ell^{\infty}\) (Q798543)

The question is the following: given \(\lim_{m}\sum^{m}_{j=1}a_ j=\lim A_ m=0\) and \(|\sum^{n}_{k=1}b_ k| =| B_ n|\leq M,\) does \[ \lim_{n,m}S_{m,n}(a,b)=\lim_{n,m}\sum^{m}_{j=1}\sum^{n}_{k=1 }ja_ jb_ k/\sqrt{j^ 2+k^ 2} \] exist? The problem can be put into a setting involving the sequence spaces \(c_ 0\) and \(\ell^{\infty}\), and so to check whether certain operator \(T:c_ 0\to\ell^ 1\) is continuous. The answer is negative. Indeed the convergence of \(S_{m,n}(a,b)\) is equivalent to that of \(L_{m,n}(a,b)=\sum^{m}_{j=1}\sum^{n}_{k=1}G_{k,j}A_ jB_ k,\) where \(G_{k,j}=F(j,k)-F(j,k+1)-F(j+1,k)+F(j+1,k+1)\) and \(F(x,y)=x/\sqrt{x^ 2+y^ 2}\) and so the answer follows from the lemma: ''Let X be a topological linear space of sequences \(A=(A_ m)\), of second category in itself. Suppose that \(\chi_ mA\to A\) in X as \(m\to\infty \), where \((\chi_ mA)_ j=A_ j\) for 1\(\leq j\leq m\), and \((\chi_ mA)_ j=0\) otherwise. Then for any matrix \((K_{k,j})\), \(k\geq 1,j\geq 1\), \(\lim_{m\wedge n\to\infty }\sum^{n}_{k=1}\sum^{m}_{J=1}K_{kj}A_ jB_ k\) exists for each \(A\in X\) and \(B\in\ell^{\infty}\) if and only if the linear operator T given by \((TA)=\sum^{\infty}_{J=1}K_{kj}A_ j k=1,2,..\). maps X continuously into \(\ell^ 1\), in which case the limit is given by \(\sum^{\infty}_{k=1}\{\sum^{\infty}_{j=1}K_{kj}A_ j\}B_ k''\).