Let \(X\), \(Y\) be two complete bornological locally convex spaces with bornologies \({\mathcal B}_ X\), \({\mathcal B}_ Y\) respectively. Let \(\mathcal U\), \(\mathcal W\) be bases of \({\mathcal B}_ X\), \({\mathcal B}_ Y\) respectively. For \(U\in{\mathcal U}\), \(L\in L(X,Y)\), the space of all continuous operators on \(X\) to \(Y\), put \(p_{U,W}(L)=\sup_{x\in U} p_ W(L(x))\), where \(p_ W\) denotes the Minkowski functional of the set \(W\in{\mathcal W}\). Let \({\mathcal L}_{U,W}=\{L\in L(X,Y): p_{U,W}(L)<\infty\}\), with \((U,W)\in{\mathcal U}\times {\mathcal W}\), and let \({\mathcal L}_{{\mathcal U}\times{\mathcal W}}\) be the collection \(\{{\mathcal L}_{U,W}: (U,W)\in {\mathcal U}\times{\mathcal W}\}\). Defining the lattice operations \(\land\), \(\lor\) and an order \(\ll\), some of the results established in this paper are:
(1) The family \({\mathcal L}_{{\mathcal U},{\mathcal W}}\) of operator spaces is a distributive lattice.
(2) \(L(X,Y)=\lim \text{ ind}_{(U,W)\in{\mathcal U}\times{\mathcal W}} {\mathcal L}_{U,W}\).
Ideals and filters in \({\mathcal L}_{{\mathcal U}\times{\mathcal W}}\) are also characterized.