Equimodular and linearity in modular spaces ✩

In this paper, the concept of equimodular is introduced. It contains several generalizations of Mazur–Ulam’s isometric theorem in modular spaces. Let X and Y be two real modular spaces and X with δ1-midpoint shrinking whose modular ρX satisfies the Δ2-condition. Assume that an operator T maps X onto Y in an δ2-{ti}-equimodular manner for all i ∈ {0} ∪N, where {ti } is a null-sequence of nonnegative reals with the property that t0 = 1, t1 1/2, and ti ti−1 2ti for i 2. Then T is affine.  2003 Elsevier Inc. All rights reserved.