Definition:Convergent Product/Normed Algebra

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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 $\ds \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if there exists $n_0\in\N$ such that:

$(1): \quad a_n$ is invertible for $n \ge n_0$

$(2): \quad$ the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$.

Definition 2: for complete algebras

Let $\struct {A, \norm{\,\cdot\,} }$ be complete.

The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if there exists $n_0 \in \N$ such that:

the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $a\in A^\times$.

Also see

• Definition:Uniform Convergence of Product