On the Borelness of the Intersection Operation

We study the intersection operation of closed linear subspaces in a separable Banach space. We show that if the ambient space is quasi-reflexive, then the intersection operation is Borel. On the other hand, if the space contains a closed subspace with a Schauder decomposition into infinitely many non-reflexive spaces, then the intersection operation is not Borel. As a corollary, for a closed subspace of a Banach space with an unconditional basis, the intersection operation of the closed linear subspaces is Borel if and only if the space is reflexive. We also consider the intersection operation of additive subgroups in an infinite-dimensional separable Banach space, and show that if this intersection operation is Borel then the space is hereditarily indecomposable. ∗ The first author acknowledges the support of his research partially by the NSFC grant 11071129 and by the NCET grant 09-0477. ∗∗ The second author acknowledges the support of his research by the US NSF grant DMS-0901853. Received September 7, 2011 and in revised form January 10, 2012