Definition:Convergent Product/Normed Algebra

Definition
Let $\mathbb K$ be a division ring with norm $\norm{\,\cdot\,}_{\mathbb K}$.

Let $\struct{ A, \norm{\,\cdot\,} }$ be an associative normed unital algebra over $\mathbb K$.

Let $\sequence{a_n}$ be a sequence in $A$.

Definition 1
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent there exists $n_0\in\N$ such that:
 * 1) $a_n$ is invertible for $n \geq n_0$
 * 2) the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$.

Definition 2: for complete algebras
Let $\struct{ A, \norm{\,\cdot\,} }$ be complete.

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent there exists $n_0\in\N$ such that:
 * the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $a\in A^\times$.

Also see

 * Equivalence of Definitions of Uniformly Convergent Product in Normed Algebra
 * Definition:Uniform Convergence of Product