Nets in a Normed Algebra

# Nets in a Normed Algebra

Definition: A Directed Set is a set $\Lambda$ with an relation $\leq$ that satisfies the three conditions:1) $\lambda \leq \lambda$ for all $\lambda \in \Lambda$ (Reflexivity).2) If $\lambda_1 \leq \lambda_2$ and $\lambda_2 \leq \lambda_3$ then $\lambda_1 \leq \lambda_3$ (Transitivity).3) For all $\lambda_1, \lambda_2 \in \Lambda$ there exists a $\lambda \in \Lambda$ such that $\lambda_1 \leq \lambda$ and $\lambda_2 \leq \lambda$ (The Existence of an Upper Bound Between Pairs in $\Lambda$). |

Definition: Let $X$ be a normed algebra. A Net in $X$ denoted $\{ x(\lambda) \}_{\lambda \in \Lambda}$ is a function of a directed set $\Lambda$ to $X$. |

Definition: Let $X$ be a normed algebra. A net $\{ x(\lambda) \}_{\lambda \in \Lambda}$ is said to Net Converge (or simply Converge) to $x \in X$ denoted $x(\lambda) \to x$ if for every open neighbourhood $U$ of $x$ there exists a $\lambda_0 \in \Lambda$ such that if $\lambda \geq \lambda_0$ then $x(\lambda) \in U$. |

*The concept of a net is not exclusive to normed algebras. Nets are defined similarly for general topological spaces. Of course, $X$ with the norm topology is a topological space!*

Note that since $X$ is a normed space and $x \in X$ then every open ball centered at $x$ is an open neighbourhood of $x$. Thus, if $\{ x(\lambda) \}_{\lambda \in \Gamma}$ net converges to $x \in X$ then for all $\epsilon > 0$ there exists a $\lambda_0 \in \Lambda$ such that if $\lambda \geq \lambda_0$ then $x(\lambda) \in B(x, \epsilon)$, or equivalently, for all $\lambda \geq \lambda_0$ we have that:

(1)\begin{align} \quad \| x - x(\lambda) \| < \epsilon \end{align}