3.1 Inner Product Spaces

Question 1

What is an Inner product space?

Answer 1

An inner product space is a pair (V, <.,.>), where V is a vector space and <.,.> is an inner product. An inner product is a function that associates a pair of vectors with a scalar, satisfying the following axioms:
1. <x, x> >= 0 (non-negativity),
2. <x, x> = 0 if and only if x = 0 (definiteness),
3. <alpha x, y> = alpha <x, y> for any scalar alpha (homogeneity),
4. <x, y> = conjugate(<y, x>) (conjugate symmetry),
5. <x + y, z> = <x, z> + <y, z> (additivity).

Question 2

Apart from the axioms of an inner product, what are some other important properties to keep in mind?

Answer 2

Some important properties include:
1. <x, 0> = <0, x> = 0 (zero vector property),
2. <x, alpha y> = conjugate(alpha) <x, y> for any scalar alpha.

Question 3

What are some examples of inner product spaces?

Answer 3

Examples of inner product spaces include:
1. The space R^n with the inner product <x, y> = the sum of (xi * yi) from i = 1 to n,
2. The space C^n with the inner product <x, y> = the sum of (xi * conjugate(yi)) from i = 1 to n.