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 $\mathfrak{A}$ be a normed algebra. A Net in $\mathfrak{A}$ is a function $m : \Lambda \to \mathfrak{A}$ where $\Lambda$ is a directed set. Nets are often denoted by $\{ m(\lambda) \}_{\lambda \in \Lambda}$.
 Definition: Let $\mathfrak{A}$ be a normed algebra. A net $\{ x(\lambda) \}_{\lambda \in \Lambda}$ in $\mathfrak{A}$ is said to Net Converge (or simply Converge) to $x \in \mathfrak{A}$ denoted $x(\lambda) \overset{\alpha} \to x$ 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, $\mathfrak{A}$ with the norm topology is a topological space!

Note that since $\mathfrak{A}$ is a normed space and $x \in \mathfrak{A}$ then every open ball centered at $x$ is an open neighbourhood of $x$. Thus, if $\{ x(\lambda) \}_{\lambda \in \Gamma}$ net converges to $x \in \mathfrak{A}$ 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}