Abstract Let (, M, ) be a measure space. We show that if a surjective map f: S₋℉ () SY between the unit spheres of real L¹ () and of an arbitrary real normed space Y satisfies align* \\|f (x) +f (y) \|, \|f (x) -f (y) \|\=\\|x+y\|, \|x-y\|\, x, y S₋℉ (), align* then there exists a phase function: S₋℉ () \-1, 1\ such that f is a surjective isometry from S₋℉ () onto SY, and furthermore, this isometry can be extended to a linear isometry on the whole space L¹ ().
Huang et al. (Wed,) studied this question.