Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica 25 (2025)

Abstract We consider ℓ p -direct sums (1 ≤ p < ∞ ) and c 0 -direct sums of countably many normed spaces and find the dual of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize smoothness and approximate smoothness, both locally and globally. These results let us answer the Chmieliński, Khurana, and Sain question raised in [4] on the existence of a non-approximately smooth normed space whose every element is smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.