Pointwise tensor products of function spaces

By means of a mixture of linear and nonlinear techniques, the paper treats quasi-Banach function spaces as _L∞$L_{\infty }$-modules and studies the basic algebraical constructions and their interactions, placing the emphasis on tensor products. Sample result: if 0→Y→X→Z→0$0\to Y\to X\to Z\to 0$ is an exact sequence of _L∞$L_{\infty }$-modules and homomorphisms, where Y and Z are function spaces, and V   is another function space, then the tensorized sequence 0→Y⊗V→X⊗V→Z⊗V→0$0\to Y\otimes V\to X\otimes V\to Z\otimes V\to 0$ is exact too.