Completeness of Lip_0(M, N) under the extensively bounded metric d_e

Determine whether, for metric spaces M and N with distinguished point 0, the space Lip_0(M, N) consisting of all Lipschitz mappings f: M → N satisfying f(0) = 0 is complete with respect to the metric d_e(T, S) = sup_{x ∈ M, x ≠ 0} d(T(x), S(x))/d(x, 0) induced from E(M, N), under the assumption that N is a complete metric space.

Background

The paper introduces extensively bounded mappings E(M, N) between metric spaces M and N with distinguished point 0, and equips E(M, N) with the metric d_e(T, S) = sup_{x ≠ 0} d(T(x), S(x))/d(x, 0). It is shown that (E(M, N), d_e) is complete whenever N is complete.

Lipschitz mappings that vanish at the distinguished point form the space Lip_0(M, N). Traditionally, Lip_0(M, N) lacks a canonical metric structure. Since Lip_0(M, N) is a subspace of E(M, N), the metric d_e can be restricted to Lip_0(M, N). The unresolved question is whether this restricted metric makes Lip_0(M, N) complete when N is complete.

References

In general, there is no specific metric defined on Lip_{0}(M,N). However, being a subspace of E(M, N), we can provide the restriction of d_e on it. But we do not know whether, in general, Lip_0(M, N) is complete with this metric or not, even assuming the completeness of N.

— A generalization of Lipschitz mappings (2410.10677 - Karn et al., 14 Oct 2024) in Remark, Section 3 (Extensively bounded mappings)