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THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 13 May 2024, pp. 1-8
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We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.) 35(1) (2024), 60–75], proving that if
$(x_n) \subset X$ and 
$(y_n) \subset Y$ are two weakly null sequences such that 
$(x_n \otimes y_n)$ converges weakly in 
$X \widehat {\otimes }_\pi Y$, then 
$(x_n \otimes y_n)$ is also weakly null.
