Let 1 ≤ p, q < ∞ be fixed, and let R = [rjk] be an infinite scalar matrix such that 1 ≤ rjk < ∞ and supj,k rjk < ∞. Let 𝓑(𝑙p, 𝑙q) be the set of all bounded linear operator from 𝑙p into 𝑙q. For a fixed Banach algebra 𝐁 with identity, we define a new vector space SRp,q(𝐁) of infinite matrices over 𝐁 and a paranorm G on SRp,q(𝐁) as follows: let $$S^R_{p,q}({\mathbf{B}})=\{A:A^{[R]}{\in}{\mathcal{B}}(l_p,l_q)\}$$ and $G(A)={\parallel}A^{[R]}{\parallel}^{\frac{1}{M}}_{p,q}$, where $A^{[R]}=[{\parallel}a_{jk}{\parallel}^{r_{jk}}]$ and M = max{1, supj,k rjk}. The existance of SRp,q(𝐁) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space.