Irregular densifiability produced by James theorem (Q380416)

A famous theorem due to \textit{R. C. James} [Stud. Math. 23, 205--216 (1964; Zbl 0113.09303)] asserts that in all nonreflexive Banach spaces there is a continuous linear functional which does not attain its sup on the closed unit ball. The effect of this result on the ``densification by weak curves'' is investigated in depth in this paper. Some new notions are introduced such as the concept of degree of densifiability of a subset which, roughly, expresses the possibility of containing \(\alpha\)-dense curves with density \(\alpha\) as small as possible.