Iterated grand and small Lebesgue spaces.

The norm of the grand Lebesgue spaces is defined through the supremum of Lebesgue norms, balanced by an infinitesimal factor. In this paper we consider the spaces defined by a norm with an analogous expression, where Lebesgue norms are replaced by grand Lebesgue norms. Without the use of interpolation theory, we prove an iteration-type theorem, and we establish that the new norm is again equivalent to the norm of grand Lebesgue spaces. We prove that the expression involved satisfy the axioms of Banach Function Spaces, and we find explicit values of the constants of the equivalence. Analogous results are proved for small Lebesgue spaces.