The Bishop-Phelps-Bollobás Theorem and Operators on Banach Spaces

The main interest of this thesis is the Bishop-Phelps-Bollobás theorem, a quantitative statement showing that the set of bounded linear functionals attaining their maximum on a closed convex set of a real Banach space is dense in the topological dual. We study this theorem from three different aspects: norm-attaining functionals, norm-attaining operators, and numerical-radius attaining operators on Banach spaces.

We begin with a preliminary discussion, tying the above theorem to two important results in infinite-dimensional optimization: Ekeland's variational principle and the Brønsted-Rockafellar principle.

Then we establish that for a Banach space X, if T : X → C₀(L) is an Asplund operator and ||T(x₀)|| ≅ ||T|| for some ||x₀|| = 1, then there is a norm-attaining Asplund operator S : X → C₀(L) and ||u₀|| = 1 with ||S(u₀)|| = ||S|| = ||T|| such that u₀ ≅ x₀ and S ≅ T. This result also yields a density theorem for weakly compact norm-attaining operators. With certain conditions on either X or a locally compact space L, we get the Bishop-Phelps-Bollobás property for (X, C₀(L)).

The investigation of a question whether norm-attaining operators on a Banach space are dense has been parallel to the study of the denseness of numerical-radius attaining operators. Thus, we introduce a new property, the Bishop-Phelps-Bollobás property for numerical radius. We provide first two examples of spaces, satisfying this property: l₁(C) and c₀(C). Along the way, another problem is addressed. Two constructive proofs of the Bishop-Phelps-Bollobás theorem in case of l₁(C) are given.