A variety of algebra is a concept like "monoid", "group" or "ring" (but not "field"), which can be axiomatized by finitary operations (e.g. multiplication, inversion) and universally quantified axioms (e.g. associativity).
In this talk, we will describe how the theorems about a variety are all encoded in the category of finitely-generated free algebras (the syntactic category). This gives rise to Lawvere semantics: the algebras of a variety are certain multiplicative presheaves on the syntactic category, and the algebra homomorphisms are natural transformations. It is easy to generalize Lawvere semantics to allow varieties with infinitary operations. Our motivating example will be a variety whose operations are infinitary linear combinations determined by bounded Radon measures.
The algebras and homomorphisms for this "variety" are "linear spaces" and bounded linear maps (maps which commute with integration). The new category of "linear spaces" we obtain is a purely algebraic setting for functional analysis where many pathologies of topological vector space theory may be avoided.