We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from 1 into Y . Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L 1 (μ)-space for a σ -finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.

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Abstract. We show that there is an infinite-dimensional vector space of differentiable functions on R, every non-zero element of which is nowhere monotone. We also show that there is a vector space of dimension 2 c of functions R → R, every non-zero element of which is everywhere surjective.

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