Summability domain of a regular matrix ($E_A=\{x\in {\ell }^{\infty }:Ax\in c\}$)

The summability domain is the inverse image of the convergent sequence space under the bounded matrix operator on the l-infinity sequence space. It is therefore a closed linear subspace. The bounded divergence for every regular summability matrix makes it proper, so it has empty interior and is a nowhere dense set. The Baire category theorem then ensures that countably many regular methods miss some common bounded sequence.