Real Inner Products

Definition

Let $(X, \R )$ be a vector space. A function $f: X \times X \to \R $ is an inner product on the vector space $(X, \R )$ if

1. $f(x,x) \geq 0$, $= 0 \iff x = 0$,
2. $f(x+y,z) = f(x,z) + f(y, z)$,
3. $f(x,y) = f(y,x)$, and
4. $f(\alpha x,y) = \alpha f(x,y)$.

Examples

$\R ^n$ with the usual inner product is an inner product space. Some authors call any finite-dimensional inner product space over the real numbers is a Euclidean vector space.

Notation

If $f: X \times X \to \R $ is an inner product we regularly denote $f(x, x)$ by $\ip{x,x}$.

Orthogonality

Two vectors in an inner product space are orthogonal if their inner product is zero. An orthogonal family of vectors in an inner product space is a family of vectors for which distinct family members are orthogonal.

A vector is normalized if its inner product with itself is one.

1. Future editions will discuss complex inner products. ↩︎