ACIT4330-Page/content/Definitions/Subnet.md

Definition

A subnet of a Nets f: I \to X is a Nets g: J \to X and a map h : J \to I such that g = f \circ h and such that \forall i \in I \exists j \in J with h(j') \geq i \; \forall j' \geq j.

Example

[!example] !Drawing 2025-02-13 11.47.33.excalidraw Sequence:

f : I = \mathbb{N} \to \mathbb{R} : f(n) = x_{n}

Subsequence:

\{ \underbrace{x_{1}}_{= x_{1}}, \, \underbrace{x_{3}}_{= x_{2}}, \, \underbrace{x_{5}}_{= x_{3}}, \, \underbrace{x_{7}}_{= x_{4}}, \, \dots \} \to 1

Definition: g : J = \mathbb{N} \to \mathbb{R} by g(j) = x'_{j} = x_{2j-1} = f(2j-1) = f \circ h(j), where h : J \to I is defined by h(j) = 2j-1.