Geometry of $\ell_p$-direct sums of normed linear spaces

TL;DR

This paper studies the geometric properties of $ ext{ell}_p$-direct sums and $c_0$-direct sums of normed spaces, focusing on duals, smoothness, and orthogonality to deepen understanding of their structure.

Contribution

It provides new characterizations of dual spaces, support functionals, smoothness, and Birkhoff-James orthogonality in $ ext{ell}_p$- and $c_0$-direct sums, including examples with unique smoothness properties.

Findings

Characterized duals of $ ext{ell}_p$- and $c_0$-direct sums.

Identified conditions for smoothness and approximate smoothness.

Described Birkhoff-James orthogonality and its symmetry in these spaces.

Abstract

We consider $\ell_p$-direct sums ($1\leq p<\infty$) and $c_0$-direct sums of countably many normed spaces and find the duals 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 obtain examples of spaces that are not approximately smooth but where every non-zero element is approximately smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.