It is well known that any $ m \times n$ matrix defines a bounded linear map $T: \mathbb{R}^n \rightarrow \mathbb{R}^m $ through the usual matrix multiplication. Is there any such result for infinte sequences. More precisely, What types of infinite matrices define bounded linear maps from the sequence space $ \ell^p $ to $\ell^q $ for $ 1 \leq p ,q\leq \infty $?
P.S: I feel something like $A=A_{i,j}$ with $\sum_{i,j \in \mathbb{Z}}|a_{i,j}|^q < \infty$ should do. Is there any well known criteria to make it a bounded linear map?
