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Inner product

An inner product satisfies the following properties for two vectors \(v\) and \(w\), and a scalar \(\alpha\):

  1. \(\langle u+v, w\rangle=\langle u,w\rangle+\langle v,w\rangle\),
  2. \(\langle\alpha v,w\rangle=\alpha\langle v,w\rangle\),
  3. \(\langle v,w\rangle=\langle w,v\rangle\), and
  4. \(\langle v,v\rangle\geq 0\) and \(\langle v,v\rangle=0\) if and only if \(v=0\).