9. Inner Products#

Let’s look at inner products and inner product space.

Definition

Let the field \(\mathbb{F}\) be \(\mathbb{R}\) or \(\mathbb{C}\), and consider the linear space \((H, \mathbb{F})\). The function:

\[ \langle\cdot,\cdot\rangle: H \times H \rightarrow \mathbb{F} \]

is called an inner product if, for each \(x, y, z \in H\) and \(\alpha \in \mathbb{F}\):

  1. \(\langle x, y+z\rangle = \langle x, y\rangle + \langle x, z\rangle\)

  2. \(\langle x, \alpha y\rangle = \alpha \langle x, y\rangle\)

  3. \(|x|^2 \equiv \langle x, x\rangle > 0\) iff \(x \neq \theta_H\)

  4. \(\langle x, y\rangle = \overline{\langle y, x\rangle}\)

Definition

A complete inner product space is known as a Hilbert space.

Note

The completeness definition here refers to the fact that every Cauchy sequence in the space must converge.

A vector \(v \in H\) that satisfies \(|v| = \langle v, v\rangle = 1\) is said to be a unit vector.

Remark

Inner products allow us to define orthogonality of vectors and discuss angles between vectors.