Inner Product Space

Let V be the vector space over the field F where F is either field of real numbers or field of complex number. An inner product space on V is a function from V⨯V into F which assigns to each ordered pairs of vectors α,β in V by a scalar (α,β) such that

1-( α, β) = (β,ᾱ)

2-(a α +b β,γ)=a(α,γ)+b(β,γ)

3-( α, α) ≥0 and (α, α)=0 ⇨ α=0 , ∀a,b∊F and α,β,γ∊V

Also the vector space V is said to be an inner product space with respect to inner product defined on it.

Note-

(1)- If F is the field of real numbers then the inner product space is called eculidean space and if F is the field of complex numbers then it is called an unitary space.

(2)-The property first in inner product space is called Conjugate Symmetry for F=C and is called real symmetry for F=R.

(3)-The property second in the definition of inner product space is called linearity.

(4)-The property (3) is called non-negativity.