The approximation of linear operators

Abstract:

Let $L(E,F)$ be the vector space of all linear maps of $E$ into $F$. Consider a subspace $G$ of $L(E,F)$ such as all continuous maps. In $G$ distinguish a subspace $H$ of maps which are to be approximated by members of a smaller subspace $N$ of $G$. Thus we always have $N \subset H \subset G \subset L(E,F)$. Then the approximation problem which we consider is to find a locally convex linear Hausdorff topology on $G$ such that $H \subset \bar N,H = \bar N$ or the completion of $N$ is $H$. In the case where $E$ and $F$ are Banach spaces, we have approximation topologies for (i) all linear operators, (ii) all the continuous linear operators, (iii) all weakly compact operators, (iv) all completely continuous operators, (v) all compact operators, and (vi) certain subclasses of the strictly singular operators. Our method is that of considering members of $L(E,F)$ as linear forms on $E \otimes F’$. Each class of linear operators is characterized as a family of linear forms. We exploit these characterizations to develop the needed topologies. Convergence on filters appears as a natural tool in doing this; indeed, in the case of linear forms we can obtain every relevant topology via convergence on filters. Particular examples give representations of weak topologies. A by-product of the main results is that Grothendieck’s approximation condition holds when we have the weak topology on a locally convex space.