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# 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.

## 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.