Applications of Hypocontinuous Bilinear Maps in Infinite-Dimensional Differential Calculus

Paradigms of bilinear maps beta: E-1 x E-2 -> F between locally convex spaces (like evaluation or composition) are not continuous, but merely hypocontinuous. We describe situations where, nonetheless, compositions of beta with Keller C-c(n)-maps (on suitable domains) are C-c(n). Our main applications concern holomorphic families of operators, and the foundations of locally convex Poisson vector spaces.