What is a product inner over a vector space V? A product inner over a vector space V is an operation that assigns to each pair of vectors u and v ...

What is a product inner over a vector space V?

A product inner over a vector space V is an operation that assigns to each pair of vectors u and v in V a real number .
A product inner over a vector space V is a function that associates a real number ‹u, v› with each pair of vectors u and v that satisfies the following axioms: i. (v, v) ≥ 0 ii. (v, v) = 0 if and only if v = 0. iii. (u, v +w) = (u, v)+ (u, w) iv. (u + v, w) = (u, w)+(v, w) v. (u, v) = (v, u) vi. (αu, v) = α(u, v) vii. (u, αv) = α(u, v)
A product inner over a vector space V is a set of orthogonal vectors.