In this short note, we prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we are able to answer a question from the literature proving that if (xₙ) ⊂ X and (yₙ) ⊂ Y are two weakly null sequences such that (xₙ ⊗ yₙ) converges weakly in X ⊗_π Y, then (xₙ ⊗ yₙ) is also weakly null.
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Colin Petitjean (2024) studied this question.
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