The set of automorphisms of B(H) is topologically reflexive in B(B(H))
The aim of this paper is to prove the statement announced in the title which can be reformulated in the following way. Let H be a separable infinite-dimensional Hilbert space and let Φ: B(H) → B(H) be a continuous linear mapping with the property that for every A ∈ B(H) there exists a sequence $(Φ_n)$ of automorphisms of B(H) (depending on A) such that $Φ(A)= lim_n Φ_n(A)$. Then Φ is an automorphism. Moreover, a similar statement holds for the set of all surjective isometries of B(H).