On the joint continuity of module multiplication

Abstract

Let A be any topological algebra over R or C. We show that the property of a topological left (right or two-sided) A-module to have a jointly continuous action of A is inherited by submodules, quotient modules, completion, direct products, direct sums, projective limits and injective limits. In the case of commutative topological A-bimodules, the same property is inherited by topological tensor products.

Keywords

• joint continuity of the module multiplication
• topological modules
• submodules
• completion
• unitization
• direct products and sums
• projective and inductive limits
• tensor product.