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.
1. Balachandran, V. K. Topological Algebras. Reprint of the 1999 original. North-Holland Mathematics Studies, Vol. 185. North- Holland Publishing Co., Amsterdam, 2000.
2. Haralampidou, M., Oudadess, M., Palacios, L. and Signoret, C. On locally A-convex modules. Mediterr. J. Math., 2022, 19(1), 23.
3. Köthe, G. Topological Vector Spaces I. Die Grundlehren der mathematischen Wissenschaften, Vol. 159. Springer, Berlin, Heidelberg, New York, 1969.