Norms, Distance, and Unit Ball
Norms
A function \(f: \mathbb{R}^n \rightarrow \mathbb{R}\) with domain \(\mathbb{R}^n\) is called a norm if:
- \(f\) is nonnegative: \(f(\mathbf{x}) \geq 0\), \(\forall \mathbf{x} \in \mathbb{R}^n\).
- \(f\) is definite: \(f(\mathbf{x}) = 0\) only if \(\mathbf{x} = \mathbf{0}\).
- \(f\) is homogeneuous: \(f( t \mathbf{x}) = |t| f(\mathbf{x})\), for all \(\mathbf{x} \in \mathbb{R}^n\) and \(t \in \mathbb{R}\).
- \(f\) satisfies the triangle inequality: \(f(\mathbf{x} + \mathbf{y}) \leq f(\mathbf{x}) + f(\mathbf{y})\), for all \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\).
Distance
A norm is a measure of the length of a vector \(\mathbf{x}\). We can measure the distance between two vectors \(\mathbf{x}\) and \(\mathbf{y}\):
\[
D(\mathbf{x}, \mathbf{y}) = ||x - y||
\]
Unit Ball
The set of all vectors with norm less than or equal to one is called unit ball of the norm \(|| \cdot ||\):
\[
B = \left\{ \mathbf{x} \in \mathbb{R}^n \ | \ ||\mathbf{x}|| \leq 1 \right\}.
\]