Duality of topological modules over normed rings (Q1637173)

The paper studies topological (left or right) modules over unital normed rings. Let \(M, N\) be two topological left modules over a unital normed ring \(R\). Then the set (actually, a topological group under the pointwise addition) of all continuous \(R\)-homomorphisms from \(M\) to \(N\) is denoted by \({\mathcal B}(M, N)\). In the paper, it is shown that, if \(M\) and \(N\) are locally bounded and \(N\) is Hausdorff, then \({\mathcal B}(X, Y)\) is a locally bounded Hausdorff space. Let \(R\) be a unital topological ring and \(S\) an admissible ring. In this case, the functors \({\mathcal B}(N, -)\) and \({\mathcal B}(-, N)\) are studied for a topological \((R, S)\)-bimodule \(N\). It is shown that, for any topological left \(R\)-module \(M\), the set \({\mathcal B}(N, M)\) is a topological left \(S\)-module and the set \({\mathcal B}(M, N)\) is a topological right \(S\)-module. Some properties of topologically (left or right) exact sequences are also proved for admissible rings.