  Formal power series products appear in nonlinear control theory when systems
modeled by Chen-Fliess series are interconnected to form new systems. In fields
like adaptive control and learning systems, the coefficients of these formal
power series are estimated sequentially with real-time data. The main goal of
the present article is to prove the continuity and analyticity of such products
with respect to several natural (locally convex) topologies on spaces of
locally convergent formal power series in order to establish foundational
properties behind these technologies. In addition, it is shown that a
transformation group central to describing the output feedback connection is in
fact an analytic Lie group in this setting with certain regularity properties.
