ABSTRACT
An inner product on set V is a function that maps ordered pairs (x, y) from V × V (that is x and y are elements of V) to a number < x, y > while satisfying the following properties:
For all v in V, < v, v > ≥ 0 and < v, v >= 0 if and only if https://www.w3.org/1998/Math/MathML"> v = 0 → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315160993/68666890-58bf-4713-8369-b2c9df48f070/content/inq_chapter4_73_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
For all u, v, and w in V, < u, v + w >=< u, v > + < u, w >.
For all u and v in V and scalar k, https://www.w3.org/1998/Math/MathML"> < k u , v > = < u , k v > = k < u , v >. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315160993/68666890-58bf-4713-8369-b2c9df48f070/content/equ_chapter4_73_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
For all u and v in V, https://www.w3.org/1998/Math/MathML"> < u , v > = < v , u > ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315160993/68666890-58bf-4713-8369-b2c9df48f070/content/inq_chapter4_73_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
