A characterization of spaces l-equivalent to the unit interval

Two topological spaces X and Y are said to be l-equivalent if there is a linear homeomorphism between Cp(X) and Cp(Y), the spaces of continuous functions on X and Y endowed with the pointwise convergence topology. We prove a necessary condition on a topological space to be l-equivalent to the n-dimensional cube. This, in particular, combined with a recent result of R. Gَrak, characterizes the topological spaces l-equivalent to the compact interval.