Definition:Induced Norm on Quotient of Cauchy Sequences

Definition

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\CC$ be the ring of Cauchy sequences over $R$

Let $\NN$ be the set of null sequences.

For all $\sequence {x_n} \in \CC$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \NN$

Let $\norm {\, \cdot \,}_1: \CC \,\big / \NN \to \R_{\ge 0}$ be defined by:

$\ds \forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = \lim_{n \mathop \to \infty} \norm {x_n}$

$\norm {\, \cdot \,}_1$ is called the induced norm on the quotient ring of Cauchy sequences.

See also

• Quotient Ring of Cauchy Sequences is Normed Division Ring where it is shown the induced norm is a norm.

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.3$ Construction of the completion of a normed field: Definition $1.20$