  In this work, I address a primary issue with adapting categorical and
algebraic concepts to functional analytic settings, the lack of free objects.
Using a "normed set" and associated categories, I describe constructions of
normed objects, which build from a set to a vector space to an algebra, and
thus parallel the natural progression found in algebraic settings. Each of
these is characterized as a left adjoint functor to a natural forgetful
functor. Further, the universal property in each case yields a "scaled-free"
mapping property, which extends previous notions of "free" normed objects.
  In subsequent papers, this scaled-free property, coupled with the associated
functorial results, will give rise to a presentation theory for Banach algebras
and other such objects, which inherits many properties and constructions from
its algebraic counterpart.
