Über kompakte Operatoren in lokalkonvexen Räumen

Let E, F be (B)-spaces and let E have the approximation property. Then the closure of FE in L_b (F,E) is identical with the space K(F,E) of compact maps in L(F,E). The following generalization of this well known statement is proved: Let E, F be complete, locally convex spaces and let E have the approximation property. Then, where denotes a 0-neighborhood base in F and refers to the completion of the tensor product with respect to the topology of bi-equicontinuous convergence.The last part is concerned with the adjoint map of a compact operator.