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 $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if 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 if and only if 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