L-Orthogonal Elements and L-Orthogonal Sequences.

The article investigates the relationship between |$L$|-orthogonal sequences and |$L$|-orthogonal elements in Banach spaces, where an |$L$|-orthogonal sequence |$\{x_n\}$| in the unit ball satisfies |$\|x + x_n\| \to 1 + \|x\|$| for every |$x$|, and an |$L$|-orthogonal element |$x^{**}$| in the bidual satisfies |$\|x + x^{**}\| = 1 + \|x\|$| for all |$x$|. The main focus is whether every |$L$|-orthogonal sequence contains |$L$|-orthogonal elements in its weak* closure. The authors provide positive results under set-theoretic assumptions: if the Banach space has density character at most the pseudointersection number |$\mathfrak{p}$|, or if limits are taken through selective ultrafilters, then such |$L$|-orthogonal elements exist. Conversely, they show that the general question is independent of the usual axioms of set theory (ZFC), exhibiting examples where |$L$|-orthogonal sequences have weak* cluster points that are not |$L$|-orthogonal, especially when ultrafilters are not |$Q$|-points. The paper also explores the abundance of |$L$|-orthogonal elements in separable octahedral Banach spaces, proving that the set of such elements can contain infinite-dimensional isometric copies of dual spaces of uniformly convex Banach spaces finitely representable in |$\ell_1$|.