Bilinear maps of locally convex spaces (Q921460)

Let u: \(X\times Y\to Z\) be a bilinear mapping where X, Y and Z are locally convex topological vector spaces. The mapping u is said to be continuous in each variable separately if the linear mappings u(x,\(\cdot): Y\to Z\) and u(\(\cdot,y): X\to Z\) are continuous for every \(x\in X\) and \(y\in Y\) respectively. If X and Y are inductive limits of the functions \(\phi_ i: X_ i\to X\), \(i\in I\), and \(\psi_ j: Y_ j\to Y\), \(j\in J\), respectively, then the bilinear mapping \(u_{ij}: X_ i\times Y_ j\to Z\) is defined by setting \(u_{ij}(\cdot,\cdot)=u(\phi_ i(\cdot),\psi_ j(\cdot)).\) The author investigates relations between the continuity properties of u and \(u_{ij}\). Among other statements he proves that u is continuous in each variable separately if and only if \(u_{ij}\) have the same property for every \(i\in I\) and \(j\in J\).