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.