3.5a Vector Norm

Vector Norm

A vector norm is a function $\Vert \cdot \Vert :\mathcal{X}\to \mathbb{R}$ that provides a way to measure the magnitude of vectors in a vector space. It satisfies the following three properties for all vectors $\vec{x},\vec{y}\in \mathcal{X}$ and scalars $\alpha \in \mathbb{R}$:

1. (Positive-Definiteness) $\Vert \vec{x}\Vert \ge 0$, where $\Vert \vec{x}\Vert =0⟺\vec{x}=\vec{0}$.
2. (Triangle Inequality) $\Vert \vec{x}+\vec{y}\Vert \le \Vert \vec{x}\Vert +\Vert \vec{y}\Vert$.
3. (Positive Homogeneity) $\Vert \alpha \cdot \vec{x}\Vert =|\alpha |\cdot \Vert \vec{x}\Vert$.