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Define an inner product.

For x,y\in S, the inner product \langle x,y\rangle is a function \langle \cdot ,\cdot \rangle :S\times S\longrightarrow {\mathbb  {C}} (or {\mathbb  {R}} if S\, is a real vector space) with the properties:

  1. \langle x,y\rangle =\overline {\langle y,x\rangle }
  2. \langle \alpha x,y\rangle =\alpha \langle x,y\rangle
  3. \langle x+y,z\rangle =\langle x,z\rangle +\langle y,z\rangle
  4. \langle x,x\rangle >0 if x\neq 0\,, and \langle x,x\rangle =0\Longleftrightarrow x=0

Main Page : Linear Algebra : Inner Products