Norm-attaining property for a dual pair of Banach spaces (Q323835)

Let \(X, Z\) be complex Banach spaces, \(Q: X \times Z \to \mathbb C\) be a duality, i.e., a bounded bilinear map. For every \(x \in X\), denote by \(\theta(x) \in Z^*\) the functional \(z \mapsto Q(x,z)\). According to the authors' definition, \(Z\) has the norm-attaining property on a convex subset \(C \subset X\) with respect to \(Q\) if for every \(z \in Z\) there is an \(x_0 \in C\) such that NEWLINE\[NEWLINE \|z\|_Z = \sup\{\mathrm{Re\;} Q(x,z): x \in C\} = \mathrm{Re\;} Q(x_0,z). NEWLINE\]NEWLINENEWLINENEWLINEThe following theorem is the main result of the paper. If \(Z\) has the norm-attaining property on a convex subset \(C \subset X\) with respect to \(Q\) and the norm-closure of \({\theta(C)}\) contains a \(\sigma\)-compact dense subset in the weak topology \(\sigma(Z^*, Z^{**})\), then the norm-closure of \({\theta(C)}\) equals \(B_{Z^*}\).NEWLINENEWLINEAs a corollary, it is demonstrated that if a Banach space \(Z\) admits a norm separable boundary \(B\), then \(Z^*\) is norm separable. Some other applications are given. A separate section is devoted to the norm-attaining property in Fourier spaces.