A survey on Lipschitz-free Banach spaces (Q2822633)

Let \(M\) be a metric space with a distinguished point, denoted by \(0\). (If the metric space is a Banach space, the canonical choice for the distinguished point is the origin.) One denotes by \(\mathrm{Lip}_0(M)\) the Banach space of all Lipschitz functions on \(M\) that vanish at \(0\). For \(x\in M\), the evaluation functional \(\delta(x): f\mapsto f(x)\) belongs to \(\mathrm{Lip}_0(M)^*\), and the closed linear span of these \(\delta(x)\) is a predual of \(\mathrm{Lip}_0(M)\), called the Lipschitz free space \(\mathcal{F}(M)\). The free space can be understood as a linearisation of \(M\), and it has the universal property to extend every Lipschitz map \(F: M\to N\) to a linear operator \(T_F: \mathcal{F}(M)\to \mathcal{F}(N)\) of the same norm. The Lipschitz free space was investigated in great detail in the seminal paper [\textit{G. Godefroy} and \textit{N. J. Kalton}, Stud. Math. 159, No. 1, 121--141 (2003; Zbl 1059.46058)], where this name was coined as well, and the paper under review surveys some of the progress made in the last 15 years, mostly by the authors of [loc. cit.] and their collaborators.NEWLINENEWLINEAn especially important case is the one in which the metric space is a Banach space. The second section contains, among many other things, a proof of the fact that every separable Banach space is isometric to a \(1\)-complemented subspace of its free space. In Section 3, the validity of the (bounded) approximation property for \(\mathcal{F}(M)\) is investigated. Section 4 studies asymptotic smoothness; one of the consequences (in fact known already before the appearance of [loc. cit.]) is that a Banach space Lipschitz isomorphic to \(c_0\) is actually linearly isomorphic to \(c_0\). Finally, Section 5 is devoted to the notion of norm attainment for Lipschitz maps, and in the last section the author presents a commented list of open problems.