An elementary, illustrative proof of the Rado–Horn theorem

Abstract

The Rado-Horn theorem provides necessary and sufficient conditions for when a family of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subfamily of the vectors satisfies the so-called RadoHorn inequality. In this paper we provide an elementary proof of the Rado-Horn theorem as well as results for the redundant case. Previous proofs give no information about how to actually partition the vectors; we use ideas present in our proof to find subfamilies of vectors which may be used to construct a kind of "optimal" partition.Published by Elsevier Inc.

Le Problème de Kadison-Singer

“…On connaît maintenant de nombreuses preuves de ce résultat, toutes hautement non triviales ; voir [18] pour une intéressante discussion. Il y a cependant une implication très facile (exercice : trouver laquelle et la démontrer).…”