On Algebras Which are Inductive Limits of Banach Spaces

We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra, and show that the algebra of holomorphic functions on a compact set is such an algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebras, and in particular, in the associated Wiener algebra.