Inner Products

An inner product is a function that generalizes the dot product. Consequently, we’ll start by defining a dot product and then move onto the inner product.

Dot Product

Definition: Inner Product

An inner product is a function $f: V \times V \rightarrow \mathbb{F}$ (where $\mathbb{F}$ is either the real numbers or the complex numbers) satisfying three properties. Here, $x, y, z \in X$ and $a, b \in \mathbb{C}$.

1. Hermitian Symmetric: $\langle x, y \rangle = \bar{\langle y, x \rangle}$

2. Conjugate Bilinear: $\langle a x + by, z \rangle = a \langle x , z \rangle + b\langle y, z \rangle$

and $\langle x, a y + b z \rangle = \bar{a} \langle x, y \rangle + \bar{b} \langle x, z \rangle$

1. Positive definite: $\langle x, x \rangle \geq 0$ and $\langle x, x \rangle = 0 \Leftrightarrow x = 0$

Inner Product Space

An inner product space is a vector space $V$ equipped with an inner product on that linear space.