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arXiv · 2404.03322

The set of elementary tensors is weakly closed in projective tensor products

Colin Petitjean
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Abstract

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 $(x_n) \subset X$ and $(y_n) \subset Y$ are two weakly null sequences such that $(x_n \otimes y_n)$ converges weakly in $X \widehat{\otimes}_πY$, then $(x_n \otimes y_n)$ is also weakly null.

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Colin Petitjean. 2024-04-04. The set of elementary tensors is weakly closed in projective tensor products. https://doi.org/10.1017/s0004972724000376

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