Inner product
An inner product satisfies the following properties for two vectors \(v\) and \(w\), and a scalar \(\alpha\):
- \(\langle u+v, w\rangle=\langle u,w\rangle+\langle v,w\rangle\),
- \(\langle\alpha v,w\rangle=\alpha\langle v,w\rangle\),
- \(\langle v,w\rangle=\langle w,v\rangle\), and
- \(\langle v,v\rangle\geq 0\) and \(\langle v,v\rangle=0\) if and only if \(v=0\).