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 VχV 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.
(5)- The
inner product space is also called as dot-product or scalar product.
Norm of a vector space:-
Let V be an
inner product space if α∊V then the norm or length of α written as ||α||
and is defined as the positive square root of (α,α) i.e. ||α||=√(α,α)
If ||α||=1 then α is called a limit
vector.
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