AI 中文总结
本文证明无额外假设时Banach格的Fremlin投射张量积保持初等张量的两类无界收敛，借助张量-格估计简化了此前需额外假设的持久性结果的证明。
AI 中文摘要
我们证明，在无任何额外假设的情况下，Banach格的Fremlin投射张量积保持初等张量的无界范数收敛和无界绝对弱收敛。主要工具是控制形如 $\lvert x_{\alpha}\otimes y_{\beta}-x\otimes y\rvert \wedge w$ 的张量-格估计，其中 $w$ 是张量积中的正元。这为之前在额外假设下得到的持久性结果提供了直接且简化的证明。
英文摘要
We prove that the Fremlin projective tensor product of Banach lattices preserves unbounded norm convergence and unbounded absolute weak convergence of elementary tensors without any additional assumptions. The main tool is a tensor-lattice estimate controlling expressions of the form \[ \lvert x_α\otimes y_β-x\otimes y\rvert \wedge w, \] where $w$ is positive in the tensor product. This provides a direct and simplified proof of permanence results previously obtained under extra hypotheses.
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