Norm

Last updated Nov 1, 2022

Let $X$ be a Vector Space over $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. A Norm $||\cdot||: X \to \mathbb{R}_{\geq 0}$ is a Function such that

1. Non-Degeneracy: $||\mathbf{x}|| = 0$ If and Only If $\mathbf{x} = 0$
2. Homogeniety: $||c \mathbf{x}|| = |c| ||x||$ for $c \in \mathbb{K}$ and $\mathbf{x} \in X$
3. Triangle Inequality: $||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}||$ for $\mathbf{x}, \mathbf{y} \in X$

An Extended Norm whose Function Image is a subset of $\mathbb{R}_{\geq0}$.