  For each positive integer k, we describe a map f from the complex plane to a
suitable non-complete complex locally convex space such that f is k times
continuously complex differentiable but not k+1 times, and hence not complex
analytic. We also describe a complex analytic map from l^1 to a suitable
complete complex locally convex space which is unbounded on each non-empty open
subset of l^1. Furthermore, we present a smooth map from the real line to a
non-complete locally convex space which is not real analytic although it is
given locally by its Taylor series around each point. As a byproduct, we find
that free locally convex spaces over subsets of the complex plane with
non-empty interior are not Mackey complete.
