A Note on Unbounded Convergences in Fremlin Projective Tensor Products

Abstract

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.

arXiv comment8 pages