Abstract
Given separable Frechet spaces, E, F, and G, let L ( E , F ) , L ( F , G ) \mathcal {L}(E,F),\mathcal {L}(F,G) , and L ( E , G ) \mathcal {L}(E,G) denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of pointwise convergence and the topology of compact convergence. We will show that if ( X , F ) (X,\mathcal {F}) is any measurable space and both A : X → L ( E , F ) A:X \to \mathcal {L}(E,F) and B : X → L ( F , G ) B:X \to \mathcal {L}(F,G) are Borelian, then the operator composition B A : X → L ( E , G ) BA:X \to \mathcal {L}(E,G) is also Borelian. Further, we will give several consequences of this result.
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