This chapter presents a survey of notions and facts needed throughout the book. As was
already mentioned in the introduction, our key references are W. Rudin [403] and M. Fabian …

A boundary for a real Banach space is a subset of the dual unit sphere with the property that
each element of the Banach space attains its norm on an element of that subset. Trivially, the …

A Banach space X is Grothendieck if the weak and the weak⁎ convergence of sequences in
the dual space X⁎ coincide. The space ℓ∞ is a classical example of a Grothendieck space …

The purpose of this paper is twofold: firstly, to provide an accessible proof of James' weak
compactness theorem that is able to be taught in a first-year graduate class in functional …

This is a survey about one of the most important achievements in optimization in Banach
space theory, namely, James' weak compactness theorem, its relatives, and its applications …

Let A and B be closed, convex and bounded subsets of a weakly sequentially complete
Banach space E which both are not weakly compact. Then there is a linear form x 0⁎∈ E⁎ …

We present some extensions of classical results that involve elements of the dual of Banach
spaces, such as Bishop–Phelp's theorem and James' compactness theorem, but restricting …

We study the (I)-envelopes of the unit balls of Banach spaces. We show, in particular, that
any nonreflexive space can be renormed in such a way that the (I)-envelope of the unit ball …

A Banach space is said to be Grothendieck if weak and weak⁎ convergent sequences in the
dual space coincide. This notion has been quantified by H. Bendová. She has proved that …

Let X be a real Banach space with the closed unit ball BX and the dual X⁎. We say that X
has the intersection property (I)(general intersection property (GI), respectively) if, for each …