A note on weakly compact subsets in the projective tensor product of Banach spaces

Let X and Y be two Banach spaces. In this short note we show that every weakly compact subset in the projective tensor product of X and Y can be written as the intersection of finite unions of sets of the form co ¯ ( K X ⊗ K Y ) , where K X and K Y are weakly compacts subsets of X and Y, respectively. If either X or Y has the Dunford–Pettis property, then any intersection of sets that are finite unions of sets of the form co ¯ ( K X ⊗ K Y ) , where K X and K Y are weakly compact sets in X and Y, respectively, is weakly compact.