Inner Product Spaces

INNER PRODUCT SPACES

Introduction :

In this chapter we shall study the vector spaces over a field of reals or a field of complex numbers. For vector space over these fields we shall introduce an idea of the length of a vector and when the field is of real numbers, we shall also be able to introduce an idea of the angle between two vectors. After that we shall see that lengths and angles may be expressed interms of a certain type of scalar valued function which is called an inner product.

Many geometric notions such as angle, length, and perpendicularity in R² and R³ may be extended to more general real and complex vector spaces. All of these ideas are related to the concept of inner product.

INNER PRODUCTS AND NORMS

(a) Inner Product spaces:

Definition :

Let V be a vector space over F. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F, denoted <x,y>, such that for all x, y and z in V and all c in F, the following axioms hold.

(d) <x,x> > 0 if x ≠ 0

Note that (c) reduces to <x,y> = <y, x> if F = R. Conditions (a) and (b) simply require that the inner product be linear in the first component.

It is easily shown that if a₁, a₂, ..., a_n Є F and y, v₁, v₂, ..., v_n Є V, then

Note: Standard inner product is usually called the dot product and is denoted by

Example 1.

Example 2.

If <x,y> is any inner product on a vector space V and r> 0, we may define another inner product by the rule <x,y>' = r <x,y>.

If r ≤ 0, then (d) would not hold.

Example 3.

Let V = C([0, 1]), the vector space of real-valued continuous functions on [0, 1].

Since the preceding integral is linear in f, (a) and (b) are immediate, and (c) is trivial. If f ≠ 0, then f² is bounded away from zero on some subinterval [0, 1] and hence 

Definition :

Example 4.

Note: If x and y are viewed as column vectors in Fⁿ, then <x,y> = y*x.

In the case that A has real entries, A* is simply the transpose of A.

Example 5.

Frobenius inner product

Note :

1. A vector space V over F endowed with a specific inner product is called an inner product space.

2. If F = C, we call V a complex inner product space, whereas if F = R, we call V a real inner product space.

3. It is clear that if V has an inner product <x, y> and W is a subspace of V, then W is also an inner product space when the same function <x,y> is restricted to the vectors x, y Є W.

4. A very important inner product space that resembles C([0, 1]) is the space H of continuous complex-valued functions defined on the interval [0, 2π] with the inner product

The reason for the constant 1/2π will become evident later.

5. Every complex-valued function f may be written as f = f₁ + i f₂, where f₁ and f₂ are real-valued functions.

Hence, 

THEOREM 1.

Let V be an inner product space. Then for x, y, z Є V and C Є F, the following statements are true.

Proof :

Problem 1.

Solution :

Problem 2.

Solution :

Problem 3.

Solution :

which is a contradiction to the condition

<x, x> > 0 if x ≠ 0

.'. Given is not an inner product on P(R)

Problem 4.

Let β be a basis for a finite-dimensional inner product space.

Prove that if <x, z> = 0 for all z Є β, then x = 0

Solution :

Problem 5.

Solution :

Proof :