Let
\mathbf{B}(X,Y)
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)
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
X
×
Y
.