Operator Topologies and Graph Convergence

Let $\mathbf{B}(X,Y)$ be the continuous linear transformations from a normed linear space X to a normed linear space Y. This article presents two general results – one for the norm topology on Y and one for the weak topology on Y – that explain how convergence of sequences in $\mathbf{B}(X,Y)$ with respect to a topology of uniform convergence on a prescribed family of norm bounded subsets of X is reflected in the bornological convergence of the associated sequence of graphs with respect to a family of subsets of the Cartesian product $X\times Y$.