Tingley's problem for a Banach space of Lipschitz functions on the closed unit interval (Research on preserver problems on Banach algebras and related topics)
We prove that every surjective isometry on the unit sphere of Lip(I) of all Lipschitz continuous functions on the closed unit interval I is extended to a surjective real linear isometry on Lip(I) with the norm ∥f∥σ = |f(0)| + ∥f′∥L∞
