ABSTRACT
Let V be a vector space over the field F, where F = ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq730.tif"/> or F = ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq731.tif"/> . An inner product on V is a function (·, ·): V × V → F such that for all u, v, w ∈ V and a, b ∈ F, the following hold:
〈v, v〉 〈v, v〉 = 0 if and only if v = 0.
〈a u + b v,w〉 = a〈v, w〉 + b〈v, w〉.
For F = ℝ : 〈 u , v 〉 = 〈 v , u 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq732.tif"/> ; For F = ℂ : 〈 u , v 〉 = 〈 v , u 〉 ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq733.tif"/> (where bar denotes complex conjugation).
