Vector Space Norm

A norm on a vector space \(V\) over \(\mathbb{F}\), where \(\mathbb{F}\) is the real or complex numbers, is a map \(\|\cdot\| : V \to \mathbb{R}\) such that for any \(\boldsymbol{u}, \boldsymbol{v}\) in \(V\) and \(\lambda\) in \(\mathbb{F}\) the following properties hold:

1. Positive Definite: \(||\boldsymbol{v}|| \geq 0\) with \(||\boldsymbol{v}|| = 0\) if and only if \(\boldsymbol{v} = \boldsymbol{0}\).
2. Preserve Scalar Multiplication: \(||\lambda \boldsymbol{v}|| = |\lambda| ||\boldsymbol{v}||\)
3. Triangle Inequality: \(||\boldsymbol{v} + \boldsymbol{u}|| \leq ||\boldsymbol{v}|| + ||\boldsymbol{u}||\).